% EJC papers *must* begin with the following two lines. \documentclass[12pt]{article} \usepackage{e-jc} \specs{P3.52}{24(3)}{2017} % Please remove all other commands that change parameters such as % margins or pagesizes. % only use standard LaTeX packages % only include packages that you actually need % we recommend these ams packages \usepackage{amsthm,amsmath,amssymb} % we recommend the graphicx package for importing figures \usepackage{graphicx} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} %\usepackage[latin1]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{graphics} %\usepackage[affil-it]{authblk} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} %\newtheorem{theorem}{Theorem}[section] %\newtheorem{proposition}[theorem]{Proposition} %\newtheorem{corollary}[theorem]{Corollary} %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{remark}[theorem]{Remark} %\newtheorem{example}[theorem]{Example} %\newtheorem{construction}[theorem]{Construction} %\newtheorem{definition}[theorem]{Definition} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \title{\bf Internally fair factorizations and internally fair holey factorizations with prescribed regularity} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Aras Erzurumluo\u{g}lu\\ \small Department of Mathematics and Statistics\\[-0.8ex] \small University of Ottawa\\[-0.8ex] \small Ottawa, ON, Canada\\ \small\tt aerzurum@uottawa.ca\\ \and Chris A. Rodger\\ \small Department of Mathematics and Statistics\\[-0.8ex] \small Auburn University\\[-0.8ex] \small Auburn, AL, United States\\ \small\tt rodgec1@auburn.edu } % \date{\dateline{submission date}{acceptance date}\\ % \small Mathematics Subject Classifications: comma separated list of % MSC codes available from http://www.ams.org/mathscinet/freeTools.html} \date{\dateline{Jun 25, 2015}{Aug 28, 2017}{Sep 8, 2017}\\ \small Mathematics Subject Classifications: 05C70 } %\date{\dateline{Apr 4, 2017}{Aug 21, 2017}{Sep 8, 2017}\\ %\small Mathematics Subject Classifications: 05E05, 20G42} \begin{document} \maketitle \begin{abstract} Let $G$ be a multipartite multigraph without loops. Then $G$ is said to be \textit{internally fair} if its edges are shared as evenly as possible among all pairs of its partite sets. An \textit{internally fair factorization} of $G$ is an edge-decomposition of $G$ into internally fair regular spanning subgraphs. A \textit{holey factor} of $G$ is a regular subgraph spanning all vertices but one partite set. An \textit{internally fair holey factorization} is an edge-decomposition of $G$ into internally fair holey factors. In this paper, we settle the existence of internally fair (respectively, internally fair holey) factorizations of the complete equipartite multigraph into factors (respectively, holey factors) with prescribed regularity. % keywords are optional %\bigskip\noindent \textbf{Keywords:} graph reconstruction %conjecture; Broglington manifold; Pipletti's classification \end{abstract} \section{Introduction} This paper addresses the existence of a general type of balance in edge-colorings of graphs that have an accompanying partition of the vertex set, namely the existence of internally fair edge-colorings. When considering different criteria for balance in edge-colorings, the goal usually is to share edges of each color as evenly as possible among certain substructures. Since such edge-colorings frequently give rise to interesting combinatorial structures which are useful as ingredients in the construction of combinatorial designs, balance in edge-colorings has been of great interest to a wide combinatorics community. Very well-known results on balance in edge-colorings include de Werra's work where he proved (see \cite{dewerra1, dewerra2, dewerra3, dewerra4}) that the edges of any bipartite graph $G$ can be colored with $k$ colors such that (i) for each pair of vertices the edges joining these vertices are shared as evenly as possible among the $k$ colors, (ii) at each vertex the incident edges are shared as evenly as possible among the $k$ colors, and (iii) overall, the edges in $E(G)$ are shared as evenly as possible among the $k$ colors. In a more general setting, Hilton \cite{hilton} proved that each graph where all degrees are even, has an edge-coloring with $k$ colors (for each $k$) with the property that (i) each color appears on an even number of incident edges at each vertex, and (ii) the number of times a color appears on incident edges at a vertex $v$ differs from the number of times another color appears on incident edges at vertex $v$ by $0$ or $2$. (This result was recently extended in \cite{evenlyequitable}, where existence results are given for edge-colorings of graphs that satisfy the aforementioned notion of balance together with other notions of balance.) The most natural graphs to consider in the setting where there is an accompanying partition of the vertex set are the complete multipartite graphs. Indeed, it is on this family of graphs that we focus in this paper. The particular notion of balance that we consider in this paper is defined next. Let $K(n,p)$ be the complete multipartite graph with $p$ parts and $n$ vertices in each part (note that $K(1,p) = K_{p}$). Let $\lambda K(n,p)$ denote the $\lambda$-fold complete multigraph, in which if two vertices are joined by $\epsilon$ edges ($\epsilon \in \mathbb{Z}_{2}$) in $K(n,p)$ then they are joined by $\lambda \epsilon$ edges in $\lambda K(n,p)$. An edge-coloring of $\lambda K(n,p)$ is said to be \textit{internally fair} if within each color class the edges are shared as evenly as possible among the pairs of parts. As described in \cite{oldpaper}, if each color class in an edge-coloring of $\lambda K(n,p)$ induces a $k$-regular spanning subgraph for the same value of $k$, then the notion of internally fair is identical to the notion of fairness in \cite{oldpaper}, where an edge-coloring is said to be \textit{fair} if the edges between each pair of parts are shared out as evenly as possible among the color classes. %$V_{x}$ and $V_{y}$ and for each pair of colors $i$ and $j$, $||c_{i}(V_{x}, V_{y})| - |c_{j}(V_{x}, V_{y})|| \leq 1$, where $c_{i}(V_{x}, V_{y})$ denotes the set of edges colored $i$ between $V_{x}$ and $V_{y}$. An \textit{$r$-factor} of $\lambda K(n,p)$ is an $r$-regular spanning subgraph of $\lambda K(n,p)$. Suppose that $R_{t}=\{r_{0}, r_{1}, \ldots, r_{t-1}\}$ is a multiset of positive integers. The graph $\lambda K(n,p)$ is said to have an \textit{$R_{t}$-factorization} if the edge set of $\lambda K(n,p)$ can be partitioned into subsets $F_{0}, \ldots, F_{t-1}$ such that for each $i \in \mathbb{Z}_{t}$, $F_{i}$ is an $r_{i}$-factor of $\lambda K(n,p)$. If $r_{i}=r$ for all $i \in \mathbb{Z}_{t}$, then $\lambda K(n,p)$ is said to have an \textit{$r$-factorization}. A 2-factorization of $\lambda K(n,p)$ where all the factors are connected is said to be a \textit{hamiltonian decomposition}. It is useful to represent factorizations of $\lambda K(n,p)$ as edge-colorings: two edges are colored the same if and only if they are in the same factor of $\lambda K(n,p)$. In this context, if $\lambda K(n,p)$ has partition $P= \{V_{0}, \ldots, V_{p-1}\}$ of the vertex set, then an internally fair $R_{t}$-factorization of $\lambda K(n,p)$ has the property that for all $p_{1}, p_{2}, p_{3}, p_{4} \in \mathbb{Z}_{p}$ with $p_{1} < p_{2}$, $p_{3} < p_{4}$ and for all $i \in \mathbb{Z}_{t}$, $||F_{i}(p_{1}, p_{2})|- |F_{i}(p_{3}, p_{4})|| \leq 1$ where for $0 \leq j < k \leq p-1$, $F_{i}(j, k)$ is the set of edges in the $r_{i}$-factor $F_{i}$ joining a vertex in $V_{j}$ to a vertex in $V_{k}$. In 2002 Leach and Rodger \cite{leach2002} completely settled the existence problem for fair hamiltonian decompositions of $K(n,p)$ (so each factor is not only $2$-regular, but also connected) showing that they exist if and only if $n(p-1)$ is even. More recently the authors \cite{oldpaper} settled the existence problem for fair 1-factorizations of $K(n,p)$ showing that they exist if and only if $np$ is even. It is not difficult to see that a $1$-factorization of $K(n,p)$ corresponds to a quasigroup of order $np$ with diagonal holes of size $n$. The discovery of holey quasigroups has allowed big breakthroughs in combinatorial design theory during the 1980's. Holey quasigroups with certain properties have been used in many instances as substitutes for complete quasigroups when the required complete quasigroups with the imposed set of properties cannot exist. For example, an easy observation shows that there are no idempotent symmetric quasigroups of even order; however, symmetric quasigroups with holes of size 2 do exist, and these structures are often usefully employed to construct certain designs. %holey structures were exactly the right way to deal %with situations where the desired property could never occur in the diagonal positions, but could be guaranteed to be satisfied elsewhere. Therefore from a design theoretic perspective, it is also of interest to consider partitions of the edges of $\lambda K(n,p)$, each element inducing a factor %of into factors of $\lambda K(n,p)-V_{i}$ for some part $V_{i}$ in the natural partition of $\lambda K(n,p)$. For each part $V_{i} \in P$, an $r$-factor of $\lambda K(n,p)- V_{i}$ is said to be a \textit{holey $r$-factor} (or \textit{holey parallel class}) of $\lambda K(n,p)-V_{i}$ ($V_{i}$ is said to be the \textit{deficiency} of the holey parallel class). A \textit{holey $r$-factorization} of $\lambda K(n,p)$ is a partition of $E(\lambda K(n,p))$ into holey parallel classes. When $n=1$, a holey $r$-factor (or a holey parallel class) is said to be a \textit{near $r$-factor} (or an \textit{almost parallel class}). The corresponding holey $r$-factorization is then said to be a \textit{near $r$-factorization} (or an \textit{almost resolvable $r$-factorization}). Suppose that $R(f_{p})= \{r_{i,j}| r_{i,j} \in \mathbb{Z}^{+}, i \in \mathbb{Z}_{p}, j \in \mathbb{Z}_{f_{p}(i)} \}$ is a multiset of positive integers where $f_{p}: \mathbb{Z}_{p} \rightarrow \mathbb{Z}^{+}$ is a function. Then $\lambda K(n,p)$ with partition $P= \{V_{0}, \ldots, V_{p-1}\}$ of the vertex set is said to have a \textit{holey $R(f_{p})$-factorization} if the edge set of $\lambda K(n,p)$ can be partitioned into subsets $F_{i,j}$ where for each $i \in \mathbb{Z}_{p}$ and for each $j \in \mathbb{Z}_{f_{p}(i)}$, $F_{i,j}$ is an $r_{i,j}$-factor of $\lambda K(n,p)-V_{i}$. If $r_{i,j}=r$ for each $i \in \mathbb{Z}_{p}$ and for each $j \in \mathbb{Z}_{f_{p}(i)}$, then $\lambda K(n,p)$ is said to have a \textit{holey $r$-factorization}. %A holey 2-factorization of $\lambda K(n,p)$ where all the holey factors are connected is said to be a \textit{holey hamiltonian decomposition}. To see the design theoretic connection, it is easy to observe that, for example, a holey $1$-factorization of $K(n,p)$ is the same as a $(2,1)$-frame of type $n^{p}$ (for a definition of frames, see for example page 261 of \cite{handbook}), and a holey hamiltonian decomposition of $K(n,p)$ is the same as an $(l,m)$-cycle frame of type $n^{p}$ where $l=n(p-1)$ is the length of each cycle and $m=1$ is the number of cycles containing each pair of vertices (for a definition of cycle frames, see \cite{cao} for example). A holey $R(f_{p})$-factorization is said to be \textit{internally fair} if for all $p_{1}, p_{2}, p_{3}, p_{4} \in \mathbb{Z}_{p}$ with $p_{1} < p_{2}$, $p_{3} < p_{4}$, for all $i \in \mathbb{Z}_{p} \setminus \{p_{1}, p_{2}, p_{3}, p_{4} \}$, and for all $j \in \mathbb{Z}_{f_{p}(i)}$, $||F_{i,j}(p_{1}, p_{2})|- |F_{i,j}(p_{3}, p_{4})|| \leq 1$ where for $0 \leq m < n \leq p-1$, $F_{i,j}(m,n)$ is the set of edges in the $r_{i,j}$-factor $F_{i,j}$ joining a vertex in $V_{m}$ to a vertex in $V_{n}$. For each holey factor $F$ with deficiency $V_{i}$ ($i\in \mathbb{Z}_{p}$), a part $V_{k}$ in $P$ is said to be \textit{permitted} for $F$ if $k \neq i$. Informally, the internally fair condition requires that the edges in each holey factor $F$ be shared as evenly as possible among the pairs of parts permitted for $F$ in $P$. In \cite{oldpaper} it was shown that if $r_{i,j}=1$ for all $i \in \mathbb{Z}_{p}$, and for all $j \in \mathbb{Z}_{f_{p}(i)}$, then this factorization can also be made fair, namely that $||F_{i_{1},j_{1}}(p_{1},p_{2})|-|F_{i_{2},j_{2}}(p_{1},p_{2})||\leq 1$ for all $i_{1}, i_{2} \in \mathbb{Z}_{p} \setminus \{p_{1}, p_{2}\}$, and for all $j_{k} \in \mathbb{Z}_{f_{p}(i_{k})}$ ($k = 1,2$). In \cite{oldpaper} the authors settled the existence problem for fair holey 1-factorizations of $K(n,p)$ showing that they exist if and only if $n(p-1)$ is even and $p\neq 2$. In \cite{fairholeyhamiltonian} a complete characterization was also found for when fair holey hamiltonian decompositions of $K(n,p)$ exist, which simultaneously settles the existence of cycle frames of type $n^{p}$ for cycles of the longest length. As one might expect, holey factorizations of $\lambda K(n,p)$ can be represented as edge-colorings as well: an edge-coloring of $\lambda K(n,p)$ is said to be a \textit{holey edge-coloring} if each color class induces a holey factor of $\lambda K(n,p)$. In this context, a holey edge-coloring of $\lambda K(n,p)$ is said to be \textit{internally fair} if the corresponding holey factorization of $\lambda K(n,p)$ is internally fair. The main aim of this paper is to extend results on fair factorizations and fair holey factorizations of $K(n,p)$ to the much more general setting where \begin{itemize} \item[(i)] color classes corresponding to the factors (or holey factors) are not required to have the same regularity, and \item[(ii)] $K(n,p)$ is generalized to the $\lambda$-fold graph $\lambda K(n,p)$. \end{itemize} Note that in this more general setting the relevant fairness notion to study is the internally fairness notion defined in this paper; the original fairness notion in \cite{oldpaper} cannot be satisfied in most cases where the regularities of the color classes are not all the same. Theorem \ref{theorem1} and Theorem \ref{theorem3} prove that such internally fair factorizations and internally fair holey factorizations exist for any prescribed regularity of the factors (or holey factors) as long as the (holey) factors satisfy some obvious necessary conditions. The main idea in proving these theorems is to apply the technique of amalgamations (which is described in the next section) to some carefully constructed edge-colorings of complete multigraphs with high multiplicity. The related edge-colorings are obtained by ordering a certain number of factors of the complete multigraph and coloring consecutive factors with the same color until the associated color class reaches the desired regularity. Let $G$ be a graph on the vertex set $\mathbb{Z}_{g}$. The \textit{difference} of the edge $e= \{ u,v\}$ in $G$, named so that $u2$ is even. There exists a set $\mathcal{F} = \lbrace F_{d}\mid d \in \mathbb{Z}_{(p-2)/2} \rbrace $ of $(p-2)/2$ almost resolvable $2$-factorizations of $2K_{p}$ such that for each vertex $v\in V(2K_{p})$, \begin{itemize} \item[(i)] the almost parallel classes in $\bigcup _{F\in \mathcal{F}}F$ with deficiency $v$ form a $2$-factorization of $K_{p-1}$. \end{itemize} \end{proposition} The next result follows directly from the constructions in the proof of Proposition 3.3 in \cite{oldpaper}. \begin{proposition}\label{prop3} Suppose $p>1$ is odd and $V=\mathbb{Z}_{p-1} \cup \{ \infty \}$. There exists a set $\mathcal{F}^{'} = \lbrace F'_{d}\mid d \in \mathbb{Z}_{(p-3)/2} \rbrace $ of $(p-3)/2$ almost resolvable $2$-factorizations of $2K_{p}$ and an almost resolvable $1$-factorization $F'_{(p-3)/2}$ of $K_{p}$ on the vertex set $V$ such that %for each vertex $v\in V=V(2K_{p})=\mathbb{Z}_{p-1}\cup \{ \infty \}$ \begin{itemize} %\item[(i)] the almost parallel classes in $\bigcup _{F\in \mathcal{F}}F$ with deficiency $v$ form a $2$-factor-\linebreak ization of $2K_{p-1}$, \item[(i)] %there exists $\mathcal{F}^{'} \subset %$\mathcal{F} = \mathcal{F}^{'} \cup $ such that $\mid \mathcal{F}^{'} \mid = (p-3)/2$ and for each vertex $v\in V$ the almost parallel classes with deficiency $v$ in $\mathcal{F}=\mathcal{F}^{'} \cup \{ F'_{(p-3)/2} \}$ are edge-disjoint (therefore each edge in $K_{p-1}$ on the vertex set $V\setminus \lbrace v \rbrace$ occurs in an almost parallel class in $\mathcal{F}$ with deficiency $v$), %\item[(ii)] %there exists $F\in \mathcal{F} \setminus \mathcal{F}^{'}$ such that for each $v\in V$ %each edge in $K_{p-1}$ on the vertex set $V\setminus \lbrace v \rbrace$ occurs in a $2$-factor in $\mathcal{F}^{'} \cup F_{(p-3)/2} $, %\item[(iii)] there exists an almost resolvable $1$-factorization $F^{1}$ of $K_{p}$ such that %for each $v \in V(2K_{p})$ %the almost parallel class with deficiency $v$ in $F^{1}$ together with the almost parallel classes in $\mathcal{F}^{'}$ with deficiency $v$ partition the edges of $K_{p-1}$ on the vertex set $V\setminus \lbrace v \rbrace$, and \item[(ii)] for each $v\neq \infty$ and for each $d \in \mathbb{Z}_{(p-3)/2}$, $F'_{d}(v)$ is a hamilton cycle on $V\setminus \{v\}$, and \item[(iii)] $F'_{(p-3)/2}(\infty) = \lbrace \lbrace i, i+(p-1)/2 \rbrace \mid i\in \mathbb{Z}_{(p-1)/2}\rbrace$, and for each $d \in \mathbb{Z}_{(p-3)/2}$ $F'_{d}(\infty) = \lbrace \lbrace i, i+d+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$. %if $v\neq \infty$, then $F_{d}(v)$ is a holey hamilton cycle with deficiency $v$ for each $d \in \mathbb{Z}_{(p-1)/2}$; $F_{d}(\infty) = \lbrace %\lbrace i, i+d+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, and $F^{1}(\infty) = \lbrace %\lbrace i, i+(p-1)/2 \rbrace \mid i\in \mathbb{Z}_{(p-1)/2}\rbrace$. (Note that for each $d \in \mathbb{Z}_{(p-1)/2}$, $F_{d}(\infty)$ is a holey $2$-factor with deficiency $\infty$ consisting of edges that are distance $d+1$ apart, and $F^{1}(\infty)$ is a holey $1$-factor with deficiency $\infty$ consisting of edges that are distance $(p-1)/2$ apart.) \end{itemize} \end{proposition} Note that by (iii) $F'_{(p-3)/2}(\infty)$ is a holey $1$-factor with deficiency $\infty$ consisting of edges that are distance $(p-1)/2$ apart, and for each $d \in \mathbb{Z}_{(p-3)/2}$ $F'_{d}(\infty)$ is a holey $2$-factor with deficiency $\infty$ consisting of edges that are distance $d+1$ apart. \hfill ($\ast$) \section{Main Results} Throughout the rest of the paper we assume that the partition of the vertices of $K(n,p)$ is naturally the parts of $K(n,p)$. \begin{theorem} \label{theorem1} There exists an internally fair $R_{t}$-factorization of $\lambda K(n,p)$ if and only if %$p \neq 2$ and \begin{itemize} \item[(i)] $\sum_{i=0}^{t-1}r_{i} = \lambda n(p-1)$, and \item[(ii)] for each $i \in \mathbb{Z}_{t}$, $r_{i}$ is even whenever $np$ is odd. \end{itemize} \end{theorem} \begin{proof} Suppose that there exists an internally fair $R_{t}$-factorization of $\lambda K(n,p)$. For each $i \in \mathbb{Z}_{t}$, $j \in \mathbb{Z}_{p}$ and each $v \in V_{j}$, the degree of $v$ in the $r_{i}$-factor of $\lambda K(n,p)$ contributes $r_{i}$ to the degree of $v$ in $\lambda K(n,p)$. The degree of $v$ in $\lambda K(n,p)$ is $\lambda n(p-1)$. So, condition (i) must hold. For each $i \in \mathbb{Z}_{t}$, an $r_{i}$-factor of $\lambda K(n,p)$ contains $npr_{i}/2$ edges. Hence $r_{i}$ must be even whenever $np$ is odd, which shows the necessity of condition (ii). The sufficiency of this set of conditions follows immediately by Proposition \ref{prop1} and Corollary \ref{amalgam1}. \end{proof} %\begin{corollary} %There exists a fair $r$-factorization of $\lambda K(n,p)$ if and only if $p \neq 2$ and $r|n(p-1)$. %\end{corollary} The following lemma gives a set of necessary conditions for an internally fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$ to exist. \begin{lemma} \label{Lemma} If there exists an internally fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$ then \begin{itemize} \item[(i)] $p \neq 2$, \item[(ii)] for each $i \in \mathbb{Z}_{p}$, $\sum_{j=0}^{f_{p}(i)-1}r_{i,j} = \lambda n$, and \item[(iii)] for each $i \in \mathbb{Z}_{p}$ and for each $j \in \mathbb{Z}_{f_{p}(i)}$, $r_{i,j}$ is even whenever $n(p-1)$ is odd. \end{itemize} \end{lemma} \begin{proof} %A vertex $v$ in $\lambda K(n,p)$ has degree $\lambda n(p-1)$. Suppose that there exists an internally fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$. There are no holey factors in $\lambda K(n,2)$, so $p \neq 2$, which shows the necessity of condition (i). %Let $v \in V(\lambda K(n,p))$. For each $i,k \in \mathbb{Z}_{p}$, each $v \in V_{i}$ and each $j \in \mathbb{Z}_{f_{p}(k)}$, the degree of $v$ in the holey $r_{k,j}$-factor of $\lambda K(n,p)$ contributes zero to the degree of $v$ in $\lambda K(n,p)$ if $k=i$ (since it has deficiency $V_{k}=V_{i}$), and is $r_{i,j}$ if $k \neq i$. Hence for each $i \in \mathbb{Z}_{p}$, if $v \in V_{i}$ then \[ deg_{\lambda K(n,p)}(v)= \sum_{ \begin{array}{c} \scriptstyle 0\leq k \leq p-1 \\ \scriptstyle k\neq i \end{array} } \sum_{j=0}^{f_{p}(k)-1}r_{k,j} = \lambda n(p-1). \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textnormal{(1)}\] Summing (1) over all $p$ we get \[\sum_{i=0}^{p-1}\sum_{ \begin{array}{c} \scriptstyle 0 \leq k \leq p-1 \\ \scriptstyle k\neq i \end{array} } \sum_{j=0}^{f_{p}(k)-1}r_{k,j} = p\lambda n(p-1). \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \textnormal{(2)} \] In (2), for each $i\in \mathbb{Z}_{p}$ and for each $j \in \mathbb{Z}_{f_{p}(i)}$, $r_{i,j}$ appears $p-1$ times in the summation on the left. So we get \[\sum_{i=0}^{p-1}\sum_{j=0}^{f_{p}(i)-1}r_{i,j} = p \lambda n.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\textnormal{(3)}\] Finally, from (1) and (3) we see that for each $i \in \mathbb{Z}_{p}$, \[\sum_{j=0}^{f_{p}(i)-1}r_{i,j} = \sum_{i=0}^{p-1}\sum_{j=0}^{f_{p}(i)-1}r_{i,j} -\sum_{\begin{array}{c} \scriptstyle 0\leq k \leq p-1 \\ \scriptstyle k\neq i \end{array} } \sum_{j=0}^{f_{p}(k)-1}r_{k,j} =\lambda n, \] which shows the necessity of condition (ii). For each $i \in \mathbb{Z}_{p}$, for each $j \in \mathbb{Z}_{f_{p}(i)}$ a holey $r_{i,j}$-factor of $\lambda K(n,p)$ contains $n(p-1)r_{i}/2$ edges. Hence $r_{i}$ must be even whenever $n(p-1)$ is odd, which shows the necessity of condition (iii). \end{proof} Now we prove that the necessary conditions in Lemma \ref{Lemma} are also sufficient. \begin{theorem} \label{theorem3} There exists an internally fair holey $R(f_{p})$-factorization of $\lambda K(n,$ $p)$ if and only if \begin{itemize} \item[(i)] $p \neq 2$, \item[(ii)] for each $i \in \mathbb{Z}_{p}$, $\sum_{j=0}^{f_{p}(i)-1}r_{i,j} = \lambda n$, and \item[(iii)] for each $i \in \mathbb{Z}_{p}$ and for each $j \in \mathbb{Z}_{f_{p}(i)}$, $r_{i,j}$ is even whenever $n(p-1)$ is odd. \end{itemize} \end{theorem} \begin{proof} The necessity of these conditions is proved in Lemma \ref{Lemma}, so we turn to a proof of the sufficiency. The result is trivial if $p=1$, so to prove the sufficiency we can assume that $p \geq 3$. First suppose that $p$ is even. Then by condition (iii), if $n$ is odd then $r_{i,j}$ is even for each $i \in \mathbb{Z}_{p}$ and for each $j \in \mathbb{Z}_{f_{p}(i)}$. By Proposition \ref{prop2}, there exist $(p-2)/2$ almost resolvable $2$-factorizations of $2K_{p}$, $F_{0}, \ldots, F_{(p-4)/2}$, on the vertex set $\mathbb{Z}_{p}$ such that for each $v \in \mathbb{Z}_{p}$ the almost parallel classes with deficiency $v$ in $F_{0}, \ldots, F_{(p-4)/2}$ form a $2$-factorization of $K_{p-1}$ on the vertex set $\mathbb{Z}_{p} \setminus \lbrace v \rbrace$. Extend this list for all $i \geq 0$ by defining $F_{i} = F_{j}$ if and only if $i \equiv j$ (modulo $(p-2)/2$). From this extended list, form a sequence $T= (T_{0}, \ldots, T_{(\lambda n^{2}p/2)-1})$ of $\lambda n^{2}p/2$ almost parallel classes of $\lambda n^{2}K_{p}$ where for $i \in \mathbb{Z}_{p}$ and for $k \in \mathbb{Z}_{\lambda n^{2}/2}$, $T_{i+kp}$ is the almost parallel class in $F_{k}$ with deficiency $i$. %For each $i \in \mathbb{Z}_{p}$ let $G(i,0)$ be the subgraph of $\lambda n^{2}K_{p}$ induced by the edges in $\bigcup_{k=0}^{(n/2)r_{i,0}-1}T_{i+kp}$ and color all edges in $G(i,0)$ with $(i,0)$; f For each $i \in \mathbb{Z}_{p}$ and for each $l \in \mathbb{Z}_{f_{p}(i)}$, let $s(i,l) = \sum_{j=0}^{l}(n/2)r_{i,j}$, and let $G(i,l)$ be the subgraph of $\lambda n^{2}K_{p}$ induced by the edges in $\bigcup_{k=s(i,l-1)}^{s(i,l)-1}T_{i+kp}$. Color all edges in $G(i,l)$ with $(i,l)$. Then for each $i \in \mathbb{Z}_{p}$, and for each $l \in \mathbb{Z}_{f_{p}(i)}$, color class $(i,l)$ consists of the $r_{i,l}n(p-1)/2$ edges of $r_{i,l}n/2$ almost parallel classes which all have the same deficiency $i\in V(\lambda n^{2} K_{p})$, and hence each color class is an $r_{i,l} n$-regular subgraph of $\lambda n^{2}K_{p}-i$. Furthermore, by condition (i) of Proposition \ref{prop2}, for each $i \in V(2K_{p})$ the almost parallel classes with deficiency $i$ in any set of $(p-2)/2$ consecutive almost resolvable 2-factorizations of $2K_{p}$ from the extended list $F_{0}, \ldots, F_{\lambda n^{2}/2-1}$ form a 2-factorization of $K_{p}-i$, which ensures that each color class induces a balanced multigraph (a multigraph $G$ is said to be \textit{balanced} if the multiplicity between any two pairs of vertices differs by at most $1$). Hence, the $r_{i,l}n(p-1)/2$ edges in each color class are shared out evenly among the $(p-1)(p-2)/2$ pairs of vertices in $\mathbb{Z}_{p} \setminus \lbrace i \rbrace $, so between any such pair of vertices there are $\lceil (r_{i,l}n(p-1)/2)/((p-1)(p-2)/2) \rceil = \lceil r_{i,l}n/(p-2)\rceil $ or $\lfloor (r_{i,l}n(p-1)/2)/((p-1)(p-2)/2) \rfloor = \lfloor r_{i,l}n/(p-2)\rfloor$ edges colored $(i,l)$. Therefore this $(\sum_{i=0}^{p-1}\sum_{l=0}^{f_{p}(l)-1}r_{i,l})$-edge-coloring is internally fair. By Corollary \ref{amalgam2} we conclude that there exists an internally fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$. Next suppose that $p$ is odd. It is important to note that $(p-1)/gcd(\{d,p-1\})$ is even when $d$ is odd. So by Lemma \ref{sternlenz}, %$G(D,p-1)$ has a 1-factorization whenever $D$ contains an odd number. ($\ast$ $\ast$) \medskip $G(D,p-1)$ has a 1-factorization whenever $D$ contains an odd number. \hfill ($\ast$ $\ast$) \medskip Throughout this case we will focus on the ordering $F'_{0}, F'_{1}, \ldots, F'_{(p-5)/2}, F'_{(p-3)/2}$ and the reverse ordering $F'_{(p-3)/2}, F'_{(p-5)/2}, \ldots, F'_{1}, F'_{0} $ of the $(p-3)/2$ almost resolvable $2$-factorizations of $2K_{p}$ in $\mathcal{F}^{'}$ and one almost resolvable $1$-factorization $F'_{(p-3)/2}$ of $K_{p}$ from Proposition \ref{prop3} on the vertex set $V=\mathbb{Z}_{p-1} \cup \{\infty\}$. In order to reduce redundancies we limit our attention to the first ordering and make a few remarks (noting that similar observations hold for the reverse ordering as well). %where for each $d \in \mathbb{Z}_{(p-3)/2}$ $F'_{d}(\infty) = \lbrace %\lbrace i, i+d+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, and $F'_{(p-3)/2}(\infty) = \lbrace %\lbrace i, i+(p-1)/2 \rbrace \mid i\in \mathbb{Z}_{(p-1)/2}\rbrace$. \medskip Notice that by ($\ast$ $\ast$), \medskip for $0 \leq i \leq (p-5)/2$ there exists a $1$-factorization of the graph consisting of the edges in $E(F'_{i}(\infty)) \cup E(F'_{i+1}(\infty))$ on the vertex set $\mathbb{Z}_{p-1}$ (into four 1-factors if $i<(p-5)/2$ and three 1-factors if $i=(p-5)/2$), and notice that $F'_{(p-3)/2}(\infty)$ is a $1$-factor on the vertex set $\mathbb{Z}_{p-1}$. \hfill ($\ast$ $\ast$ $\ast$) \medskip Also notice that by (ii) of Proposition \ref{prop3}, \medskip for each $v\neq \infty$ and for each $d \in \mathbb{Z}_{(p-3)/2}$, $F'_{d}(v)$ is a hamilton cycle on $V\setminus \{v\}$, and since $|V\setminus \{v\}|$ is even there exists a 1-factorization of each of these hamilton cycles into two 1-factors on the vertex set $V \setminus \{v \}$, each 1-factor being formed by taking every second edge on the hamilton cycle. \hfill ($\ast$ $\ast$ $\ast$ $\ast$) \medskip As in the previous case where $p$ is even, the plan is to proceed by finding a certain coloring of $\lambda n^{2}K_{p}$ and then applying Corollary \ref{amalgam2} to get the required internally fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$. Let $\lambda n^{2} = s(p-2)+t$ where $0\leq t \leq p-3$. We will consider eight cases depending on both the value of $t$ (modulo 4) and whether $p \equiv 1$ or $3$ (modulo 4). A rough indication as to why these subcases arise is as follows. The critical problem to address is ensuring that each color class induces a balanced multigraph. For each vertex $v$ we select $s$ near 1-factorizations of $K_{p}$ with deficiency $v$, and are then left to find $t$ more near 1-factors with deficiency $v$. This is easy to do in all cases except when $v = \infty$, where we regularly need to join pairs of near 2-factors in order to construct the near 1-factors, as described in ($\ast$ $\ast$ $\ast$). Dealing with this gives rise to the eight situations which are considered to be two cases, each with four subcases. Let $O = O_{0}, \ldots, O_{(p-3)/2}$ be an arbitrary ordering of the list $L = F'_{0}, F'_{1}, \ldots, F'_{(p-5)/2}, F'_{(p-3)/2}$. Extend the list $O$ to a list $O^*=O^*_0, O^*_1, \ldots, O^*_{s(p-1)/2-1}$ of $s(p-3)/2$ almost resolvable $2$-factorizations and $s$ almost resolvable $1$-factorizations by defining $O^*_i=O_j$ if $i \equiv j$ (modulo $(p-1)/2$). Additionally, define $t/2$ (if $t$ is even) or $(t+1)/2$ (if $t$ is odd) almost resolvable factorizations as follows: If $t$ is even, then $O^*_{s(p-1)/2}=F'_0, O^*_{s(p-1)/2+1}=F'_1, \ldots, O^*_{s(p-1)/2+t/2-1}=F'_{t/2-1}$. If $t$ is odd, then $O^*_{s(p-1)/2}=F'_{(p-3)/2}, O^*_{s(p-1)/2+1}=F'_{(p-5)/2}, \ldots, O^*_{s(p-1)/2+(t-3)/2}= \linebreak F'_{(p-t)/2}, O^*_{s(p-1)/2+(t-1)/2}= F'_{(p-t-2)/2}$. %Extend the list $O$ to a list of $s(p-3)/2 + (t/2)$ almost resolvable 2-factorizations and $s$ almost resolvable 1-factorizations by defining $O_{i}=O_{j}$ if $i \equiv j$ (modulo $(p-1)/2$). For each $v \in \mathbb{Z}_{p-1}\cup \{ \infty \}$ form a sequence $T(v)=(T(v,0), T(v,1), \ldots, T(v,q))$ (where $q=(s(p-1)+t)/2-1$ if $t$ is even, and $q=(s(p-1)+t-1)/2$ if $t$ is odd) %$T(v) =(T(v,0), T(v,1), \ldots, T(v,((s(p-1)+t)/2)-1))$ of almost parallel classes of $K_{p}$, by defining $T(v,k)$ for each $v \in \mathbb{Z}_{p-1}\cup \{\infty\}$ and for each $k \in \mathbb{Z}_{q+1}$ to be the almost parallel class (a near $1$-factor or a near $2$-factor) in $O^*_{k}$ with deficiency $v$. Now for each $v \in \mathbb{Z}_{p-1} \cup \{ \infty \}$ form a longer new sequence $W(v) = (W(v)_{0}, \linebreak W(v)_{1}, \ldots )$ from $T(v)$ as follows. First suppose $v \in \mathbb{Z}_{p-1}$. For each $v \in \mathbb{Z}_{p-1}$ and for $0 \leq k \leq (s(p-1) + t)/2-1$, if $O^*_{k} \neq F'_{(p-3)/2}$ then replace $T(v,k)$ in $T(v)$ with the two near 1-factors with deficiency $v$, say $W_{v}'(k)$ and $W_{v}''(k)$, formed from $T(v,k)$ (using ($\ast$ $\ast$ $\ast$ $\ast$)). (Note that for $v \in \mathbb{Z}_{p-1}$ if $O^*_{k} = F'_{(p-3)/2}$, then $T(v,k)$ is already a near 1-factor with deficiency $v$, which we also name with $W_{v}(k)$.) We now use two specific orderings of the list $L$ to define $W(\infty)$ depending on the parity of $t$. \medskip \textbf{Case 1}: Suppose $t$ is even (modulo 4). \medskip Let $O$ be the ordering $F'_{0}, F'_{1}, \ldots, F'_{(p-5)/2}, F'_{(p-3)/2}$. \medskip \textbf{Subcase 1.1}: Suppose $t \equiv 0$ (modulo 4) and $p \equiv 1$ (modulo 4). \medskip For each $k$ such that $k$ is even (modulo $(p-1)/2$) and $0\leq k \leq (p-9)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). For $k \equiv (p-5)/2$ (modulo $(p-1)/2$) replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). See Example \ref{example1}. \medskip \textbf{Subcase 1.2}: Suppose $t \equiv 2$ (modulo 4) and $p \equiv 1$ (modulo 4). \medskip For each $k \equiv 0$ (modulo $(p-1)/2$) replace $T(\infty, k)$ with the two near 1-factors with deficiency $\infty$, say $W'_{\infty}(k)$ and $W''_{\infty}(k)$, formed from $T(\infty,k)$ (notice that for each $k \equiv 0$ (modulo $(p-1)/2$) $T(\infty,k)= \lbrace \lbrace i, i+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, which can be factored into two near 1-factors on $\mathbb{Z}_{p-1}$ using ($\ast$ $\ast$)). For each $k$ such that $k$ is odd (modulo $(p-1)/2$) and $1\leq k \leq (p-7)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). Note that for $k \equiv (p-3)/2$ (modulo $(p-1)/2$) $T(\infty,k)$ is already a near 1-factor with deficiency $\infty$, which we also name with $W_{\infty}(k)$. See Example \ref{example2}. \medskip \textbf{Subcase 1.3}: Suppose $t \equiv 0$ (modulo 4) and $p \equiv 3$ (modulo 4). \medskip For each $k$ such that $k$ is even (modulo $(p-1)/2$) and $0\leq k \leq (p-7)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). (Note that for $k \equiv (p-3)/2$ (modulo $(p-1)/2$) $T(\infty,k)$ is already a near 1-factor with deficiency $\infty$, which we also name with $W_{\infty}(k)$.) %replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ by ($\ast$ $\ast$ $\ast$). See Example \ref{example3}. \medskip \textbf{Subcase 1.4}: Suppose $t \equiv 2$ (modulo 4) and $p \equiv 3$ (modulo 4). \medskip For each $k \equiv 0$ (modulo $(p-1)/2$) replace $T(\infty, k)$ with the two near 1-factors with deficiency $\infty$, say $W'_{\infty}(k)$ and $W''_{\infty}(k)$, formed from $T(\infty,k)$ (notice that for each $k \equiv 0$ (modulo $(p-1)/2$) $T(\infty,k)= \lbrace \lbrace i, i+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, which can be factored into two near 1-factors on $\mathbb{Z}_{p-1}$ using ($\ast$ $\ast$)). For each $k$ such that $k$ is odd (modulo $(p-1)/2$) and $1\leq k \leq (p-9)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). Finally, for $k \equiv (p-5)/2$ (modulo $(p-1)/2$) replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). See Example \ref{example4}. %Now for each $v \in \mathbb{Z}_{p-1} \cup \{\infty \}$ and for each $l \in \mathbb{Z}_{f_{p}(v)}$, let $s(v,l) = \sum_{j=0}^{l}r_{v,j}n$, and let $G(v,l)$ be the subgraph of $\lambda n^{2}K_{p}$ induced by the edges in \linebreak $\bigcup_{z=s(v,l-1)}^{s(v,l)-1}W(v)_{z}$. Color all edges in $G(v,l)$ with $(v,l)$. Then for each $v \in \mathbb{Z}_{p-1}\cup \{\infty \}$, and for each $l \in \mathbb{Z}_{f_{p}(v)}$, color class $(v,l)$ consists of the $r_{v,l}n(p-1)/2$ edges of $r_{v,l}n$ near 1-factors which all have the same deficiency $v$, and hence each color class is an $r_{v,l} n$-regular subgraph of $\lambda n^{2}K_{p}-v$. Furthermore, by condition (i) of Proposition \ref{prop3}, each color class $(v,l)$ is balanced. Hence, the $r_{v,l}n(p-1)/2$ edges in each color class are shared out evenly among the $(p-1)(p-2)/2$ pairs of vertices in $\mathbb{Z}_{p-1} \cup \{\infty \} \setminus \lbrace v \rbrace $, so between any such pair of vertices there are $\lceil (r_{v,l}n(p-1)/2)/((p-1)(p-2)/2) \rceil = \lceil r_{v,l}n/(p-2)\rceil $ or $\lfloor (r_{v,l}n(p-1)/2)/((p-1)(p-2)/2) \rfloor = \lfloor r_{v,l}n/(p-2)\rfloor$ edges colored $(v,l)$. Therefore this edge-coloring is fair. By Corollary \ref{amalgam2} we conclude that there exists a fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$. \medskip \textbf{Case 2}: Suppose $t$ is odd (modulo 4). \medskip Let $O$ be the ordering $F'_{(p-3)/2}, F'_{(p-5)/2}, \ldots, F'_{1}, F'_{0}$. \medskip \textbf{Subcase 2.1}: Suppose $t \equiv 1$ (modulo 4) and $p \equiv 1$ (modulo 4). \medskip Note that for $k \equiv 0$ (modulo $(p-1)/2$) $T(\infty,k)$ is already a near 1-factor with deficiency $\infty$, which we now name with $W_{\infty}(k)$. For each $k$ such that $k$ is odd (modulo $(p-1)/2$) and $1\leq k \leq (p-7)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). Finally for each $k \equiv (p-3)/2$ (modulo $(p-1)/2$) replace $T(\infty, k)$ with the two near 1-factors with deficiency $\infty$, say $W'_{\infty}(k)$ and $W''_{\infty}(k)$, formed from $T(\infty,k)$ (notice that for each $k \equiv (p-3)/2$ (modulo $(p-1)/2$) $T(\infty,k)= \lbrace \lbrace i, i+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, which can be factored into two near 1-factors on $\mathbb{Z}_{p-1}$ using ($\ast$ $\ast$)). See Example \ref{example5}. \medskip \textbf{Subcase 2.2}: Suppose $t \equiv 3$ (modulo 4) and $p \equiv 1$ (modulo 4). \medskip For $k \equiv 0$ (modulo $(p-1)/2$) replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). For each $k$ such that $k$ is even (modulo $(p-1)/2$) and $2\leq k \leq (p-5)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). See Example \ref{example6}. \medskip \textbf{Subcase 2.3}: Suppose $t \equiv 1$ (modulo 4) and $p \equiv 3$ (modulo 4). \medskip %For each $k \equiv 0$ (modulo $(p-1)/2$) replace $T(\infty, k)$ with the two 1-factors with deficiency $\infty$, say $W'_{\infty}(k)$ and $W''_{\infty}(k)$, formed from $T(\infty,k)$ (notice that for each $k \equiv 0$ (modulo $(p-1)/2$) $T(\infty,k)= \lbrace %\lbrace i, i+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, which can be factored into two 1-factors on $\mathbb{Z}_{p-1}$ by ($\ast$ $\ast$)). For each $k$ such that $k$ is odd (modulo $(p-1)/2$) and $1\leq k \leq (p-9)/2$, replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ by ($\ast$ $\ast$ $\ast$). Finally, for $k \equiv (p-5)/2$ (modulo $(p-1)/2$) replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ by ($\ast$ $\ast$ $\ast$). See Example \ref{example6}. %For $k \equiv 0$ (modulo $(p-1)/2$) replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ by ($\ast$ $\ast$ $\ast$). Note that for $k \equiv 0$ (modulo $(p-1)/2$) $T(\infty,k)$ is already a near 1-factor with deficiency $\infty$, which we now name with $W_{\infty}(k)$. For each $k$ such that $k$ is odd (modulo $(p-1)/2$) and $1\leq k \leq (p-5)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). %Finally for each $k \equiv (p-3)/2$ (modulo $(p-1)/2$) replace $T(\infty, k)$ with the two 1-factors with deficiency $\infty$, say $W'_{\infty}(k)$ and $W''_{\infty}(k)$, formed from $T(\infty,k)$ (notice that for each $k \equiv (p-3)/2$ (modulo $(p-1)/2$) $T(\infty,k)= \lbrace %\lbrace i, i+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, which can be factored into two 1-factors on $\mathbb{Z}_{p-1}$ by ($\ast$ $\ast$)). See Example \ref{example7}. \medskip \textbf{Subcase 2.4}: Suppose $t \equiv 3$ (modulo 4) and $p \equiv 3$ (modulo 4). \medskip For $k \equiv 0$ (modulo $(p-1)/2$) replace $T(\infty,k)$ and $T(\infty,k+1)$ with the three near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1)$ and $W'''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). For each $k$ such that $k$ is even (modulo $(p-1)/2$) and $2\leq k \leq (p-7)/2$ (modulo $(p-1)/2$), replace $T(\infty,k)$ and $T(\infty,k+1)$ with the four near 1-factors with deficiency $\infty$, say $W'_{\infty}(k, k+1), W''_{\infty}(k, k+1), W'''_{\infty}(k, k+1)$ and $W''''_{\infty}(k, k+1)$, formed from $T(\infty,k)$ and $T(\infty,k+1)$ using ($\ast$ $\ast$ $\ast$). Finally for each $k \equiv (p-3)/2$ (modulo $(p-1)/2$) replace $T(\infty, k)$ with the two near 1-factors with deficiency $\infty$, say $W'_{\infty}(k)$ and $W''_{\infty}(k)$, formed from $T(\infty,k)$ (notice that for each $k \equiv (p-3)/2$ (modulo $(p-1)/2$) $T(\infty,k)= \lbrace \lbrace i, i+1 \rbrace \mid i\in \mathbb{Z}_{p-1}\rbrace$, which can be factored into two near 1-factors on $\mathbb{Z}_{p-1}$ using ($\ast$ $\ast$)). See Example \ref{example8}. %\begin{corollary} %There exists a fair holey $r$-factorization of $K(n,p)$ if and only if $p \neq 2$ and $r|n$. %\end{corollary} \bigskip $W(v)$ has been defined for all $v \in \mathbb{Z}_{p-1} \cup \lbrace \infty \rbrace$ in all possible cases. This is used to define the color classes as follows. For each $v \in \mathbb{Z}_{p-1} \cup \{\infty \}$ and for each $l \in \mathbb{Z}_{f_{p}(v)}$, let $s(v,l) = \sum_{j=0}^{l}r_{v,j}n$, and let $G(v,l)$ be the subgraph of $\lambda n^{2}K_{p}$ induced by the edges in $\bigcup_{z=s(v,l-1)}^{s(v,l)-1}W(v)_{z}$. Color all edges in $G(v,l)$ with $(v,l)$. Then for each $v \in \mathbb{Z}_{p-1}\cup \{\infty \}$, and for each $l \in \mathbb{Z}_{f_{p}(v)}$, color class $(v,l)$ consists of the $r_{v,l}n(p-1)/2$ edges of $r_{v,l}n$ near 1-factors which all have the same deficiency $v$, and hence each color class is an $r_{v,l} n$-regular subgraph of $\lambda n^{2}K_{p}-v$. Furthermore, by condition (i) of Proposition \ref{prop3}, each color class $(v,l)$ is balanced. Hence, the $r_{v,l}n(p-1)/2$ edges in each color class are shared out evenly among the $(p-1)(p-2)/2$ pairs of vertices in $\mathbb{Z}_{p-1} \cup \{\infty \} \setminus \lbrace v \rbrace $, so between any such pair of vertices there are $\lceil (r_{v,l}n(p-1)/2)/((p-1)(p-2)/2) \rceil = \lceil r_{v,l}n/(p-2)\rceil $ or $\lfloor (r_{v,l}n(p-1)/2)/((p-1)(p-2)/2) \rfloor = \lfloor r_{v,l}n/(p-2)\rfloor$ edges colored $(v,l)$. Therefore this edge-coloring is internally fair. By Corollary \ref{amalgam2} we conclude that there exists an internally fair holey $R(f_{p})$-factorization of $\lambda K(n,p)$. \end{proof} As was suggested in the last proof, examples are now provided of the orderings that are a crucial part of the proof of Theorem \ref{theorem3}. \begin{example} \label{example1} If $p=13$ and $\lambda n^{2} = 26$, then the ordering of the $26$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{25}) = (W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), \\ W'_{i}(3), W''_{i}(3), W'_{i}(4), W''_{i}(4), W_{i}(5), W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3),\\ W''_{i}(3), W'_{i}(4), W''_{i}(4), W_{i}(5), W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1))$. The ordering of the $26$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{25}) = (W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1),\\ W''''_{\infty}(0, 1), W'_{\infty}(2, 3), W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W'_{\infty}(4, 5), W''_{\infty}(4, 5), W'''_{\infty}(4, 5), \\ W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W''''_{\infty}(0, 1), W'_{\infty}(2, 3), W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), \\ W'_{\infty}(4, 5), W''_{\infty}(4, 5), W'''_{\infty}(4, 5), W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W''''_{\infty}(0, 1))$. \end{example} \begin{example} \label{example2} If $p=13$ and $\lambda n^{2} = 24$, then the ordering of the $24$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{23}) = (W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), \\ W'_{i}(3), W''_{i}(3), W'_{i}(4), W''_{i}(4), W_{i}(5), W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), \\ W''_{i}(3), W'_{i}(4), W''_{i}(4), W_{i}(5), W'_{i}(0), W''_{i}(0))$. The ordering of the $24$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{23}) = (W'_{\infty}(0), W''_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), \\ W'''_{\infty}(1, 2), W''''_{\infty}(1, 2), W'_{\infty}$ $(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W''''_{\infty}(3, 4), W_{\infty}(5), W'_{\infty}(0), W''_{\infty}(0), \\ W'_{\infty}(1, 2), W''_{\infty}(1, 2), W'''_{\infty}(1, 2), W''''_{\infty}(1, 2), W'_{\infty}$ $(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W''''_{\infty}(3, 4), \\ W_{\infty}(5), W'_{\infty}(0), W''_{\infty}(0))$. \end{example} \begin{example} \label{example3} If $p=11$ and $\lambda n^{2} = 22$, then the ordering of the $22$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{21}) = (W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), \\ W'_{i}(3), W''_{i}(3), W_{i}(4), W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), W''_{i}(3), W_{i}(4), \\ W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1))$. The ordering of the $22$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{21}) = (W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1),\\ W''''_{\infty}(0, 1), W'_{\infty}(2, 3), W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W_{\infty}(4), W'_{\infty}(0, 1), W''_{\infty}(0, 1),\\ W'''_{\infty}(0, 1), W''''_{\infty}(0, 1), W'_{\infty}(2, 3), W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W_{\infty}(4), W'_{\infty}(0, 1), \\ W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W''''_{\infty}(0, 1))$. \end{example} \begin{example} \label{example4} If $p=11$ and $\lambda n^{2} = 20$, then the ordering of the $20$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{19}) = (W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), \\ W'_{i}(3), W''_{i}(3), W_{i}(4), W'_{i}(0), W''_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), W''_{i}(3), W_{i}(4), \\ W'_{i}(0), W''_{i}(0))$. The ordering of the $20$ near $1$-factors with deficiency $\infty$ is as follows: \\ $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{19}) = (W'_{\infty}(0), W''_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), \\ W'''_{\infty}(1, 2), W''''_{\infty}(1, 2), W'_{\infty}$ $(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W'_{\infty}(0), W''_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), \\ W'''_{\infty}$ $(1, 2), W''''_{\infty}(1, 2), W'_{\infty}(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W'_{\infty}(0),$ $W''_{\infty}(0))$. \end{example} \begin{example} \label{example5} If $p=13$ and $\lambda n^{2} = 27$, then the ordering of the $27$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{26}) = (W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), \\ W''_{i}(3), W'_{i}(4), W''_{i}(4), W'_{i}(5), W''_{i}(5), W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), W''_{i}(3), \\ W'_{i}(4), W''_{i}(4), W'_{i}(5), W''_{i}(5), W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2))$. The ordering of the $27$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{26}) = (W_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), W'''_{\infty}(1, 2), \\ W''''_{\infty}(1, 2), W'_{\infty}(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W''''_{\infty}(3, 4), W'_{\infty}(5), W''_{\infty}(5), W_{\infty}(0), W'_{\infty}(1, 2), \\ W''_{\infty}(1, 2), W'''_{\infty}(1, 2), W''''_{\infty}(1, 2), W'_{\infty}(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W''''_{\infty}(3, 4), W'_{\infty}(5), \\ W''_{\infty}(5), W_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), W'''_{\infty}(1, 2), W''''_{\infty}(1, 2))$. \end{example} \begin{example} \label{example6} If $p=13$ and $\lambda n^{2} = 25$, then the ordering of the $25$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{24}) = (W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), \\ W''_{i}(3), W'_{i}(4), W''_{i}(4), W'_{i}(5), W''_{i}(5), W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), W''_{i}(3), \\ W'_{i}(4), W''_{i}(4), W'_{i}(5), W''_{i}(5), W_{i}(0), W'_{i}(1), W''_{i}(1))$. The ordering of the $25$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{24}) = (W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W'_{\infty}(2, 3), \\ W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W'_{\infty}(4, 5), W''_{\infty}(4, 5), W'''_{\infty}(4, 5), W''''_{\infty}(4, 5), W'_{\infty}(0, 1), \\ W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W'_{\infty}(2, 3), W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W'_{\infty}(4, 5), W''_{\infty}(4, 5), \\ W'''_{\infty}(4, 5), W''''_{\infty}(4, 5), W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1))$. \end{example} \begin{example} \label{example7} If $p=11$ and $\lambda n^{2} = 23$, then the ordering of the $23$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{22}) = (W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), \\ W''_{i}(3), W'_{i}(4), W''_{i}(4), W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), W''_{i}(3), W'_{i}(4), W''_{i}(4), \\ W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2))$. The ordering of the $23$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{22}) = (W_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), W'''_{\infty}(1, 2), \\ W''''_{\infty}(1, 2), W'_{\infty}(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W''''_{\infty}(3, 4), W_{\infty}(0), W'_{\infty}(1, 2), W''_{\infty}(1, 2), \\ W'''_{\infty}(1, 2), W''''_{\infty}(1, 2), W'_{\infty}(3, 4), W''_{\infty}(3, 4), W'''_{\infty}(3, 4), W''''_{\infty}(3, 4), W_{\infty}(0), W'_{\infty}(1, 2),\\ W''_{\infty}(1, 2), W'''_{\infty}(1, 2), W''''_{\infty}(1, 2))$. \end{example} \begin{example} \label{example8} If $p=11$ and $\lambda n^{2} = 21$, then the ordering of the $21$ near $1$-factors with deficiency $i \in \mathbb{Z}_{p-1}$ is as follows: $W(i) = (W(i)_{0}, W(i)_{1}, \ldots, W(i)_{20}) = (W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3),\\ W''_{i}(3), W'_{i}(4), W''_{i}(4), W_{i}(0), W'_{i}(1), W''_{i}(1), W'_{i}(2), W''_{i}(2), W'_{i}(3), W''_{i}(3), W'_{i}(4), W''_{i}(4), \\ W_{i}(0), W'_{i}(1), W''_{i}(1))$. The ordering of the $21$ near $1$-factors with deficiency $\infty$ is as follows: $W(\infty) = (W(\infty)_{0}, W(\infty)_{1}, \ldots, W(\infty)_{20}) = (W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W'_{\infty}(2, 3), \\ W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W'_{\infty}(4), W''_{\infty}(4), W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1), W'_{\infty}(2, 3), \\ W''_{\infty}(2, 3), W'''_{\infty}(2, 3), W''''_{\infty}(2, 3), W'_{\infty}(4), W''_{\infty}(4),W'_{\infty}(0, 1), W''_{\infty}(0, 1), W'''_{\infty}(0, 1))$. \end{example} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection*{Acknowledgements} The authors wish to thank Amin Bahmanian for suggesting the problem. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \bibliographystyle{plain} % \bibliography{myBibFile} % If you use BibTeX to create a bibliography % then copy and past the contents of your .bbl file into your .tex file \begin{thebibliography}{99} \bibitem{bahmanian} M. 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