% EJC papers *must* begin with the following two lines.
\documentclass[12pt]{article}
\usepackage{e-jc}
\specs{P3.42}{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{pictex}

%\usepackage{prepictex, pictex, postpictex}



%\input prepictex
%\input pictex
%\input postpictex





% use this command to create hyperlinks (optional and recommended)
\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}

% 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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% if needed include a line break (\\) at an appropriate place in the title

\title{\bf Connected even factors in the square\\ of essentially 2-edge-connected graph}

% 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{Jan Ekstein\thanks{Supported by (i) the European Regional Development Fund (ERDF), Project, NTIS - New Technologies for Information Society,
European Centre of Excellence, CZ.1.05/1.1.00/02.0090; (ii) project GA14-19503S of the Grant Agency of the Czech 
Republic.}\\
\small University of West Bohemia, \\[-0.8ex]
\small Pilsen, Czech Republic  \\
\small\tt ekstein@kma.zcu.cz\\
\and
Baoyindureng Wu \thanks{ Supported by NSFC (No.11161046) and by Xinjiang Talent Youth Project (No.2013721012).}\\
\small Xinjiang University, \\[-0.8ex]
\small Urumgi, Xinjiang, P.R.China \\
\small\tt baoyin@xju.edu.cn\\
\and Liming Xiong \thanks{Supported by NSFC (No.11471037 and No.11671037).}\\
\small School of Mathematics and Statistics \\[-0.8ex]
\small Beijing Institute of Technology \\[-0.8ex]
\small Beijing, P. R. China\\
\small \tt lmxiong@bit.edu.cn
}

% \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{Aug 5, 2015} {Aug 22, 2017}{Sep 8, 2017}\\
\small Mathematics Subject Classifications: 05C38, 05C48}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
% should be informative, clear, and as complete as possible. Phrases
% like "we investigate..." or "we study..." should be kept to a minimum
% in favor of "we prove that..."  or "we show that...".  Do not
% include equation numbers, unexpanded citations (such as "[23]"), or
% any other references to things in the paper that are not defined in
% the abstract. The abstract will be distributed without the rest of the
% paper so it must be entirely self-contained.

\begin{abstract}
  An essentially $k$-edge connected graph $G$ is a connected graph such that deleting less than $k$ edges from $G$ cannot result in two nontrivial components.
 In this paper we prove that if an essentially 2-edge-connected graph $G$ satisfies that for any pair of leaves at distance 4 in $G$ there exists another
 leaf of $G$ that has distance 2 to one of them, then the square $G^2$ has a connected even factor with maximum degree at most 4. Moreover we show that,
 in general, the square of essentially 2-edge-connected graph does not contain a connected even factor with bounded maximum degree.
% keywords are optional

  \bigskip\noindent \textbf{Keywords:} connected even factors; (essentially) 2-edge connected graphs; square of graphs
  %}
\end{abstract}

\section {Introduction}
We consider only finite undirected simple graphs. For terminology and notation not defined in this paper we refer to \cite{WES}. Let $G$ be a connected graph.
For vertices $x, y$ of $G$, let $N_G(x)$ denote the \emph{neighborhood} of $x$ in $G$, $d_G(x)=|N_G(x)|$ the \emph{degree} of $x$ in $G$, and
$\mbox{dist}_G(x,y)$ the \emph{distance} between $x, y$ in $G$. The \emph{square} of a graph $G$, denoted by $G^2$, is the graph with same vertex set as $G$
in which two vertices are adjacent if their distance in $G$ is at most 2. Thus $G\subseteq G^2$. There are several papers (e.g. see \cite{EKS}, \cite{EL},
\cite{FAU}, \cite{FLE}, \cite{GOU}, \cite{HEN}, \cite{CHA}, and \cite{CHI}) about hamiltonian properties in the square of a graph. This paper deals with
connected even factors which generalize some previous known results.

A \emph{factor} in a graph $G$ is a spanning subgraph of $G$. A \emph{connected even factor} in $G$ is a connected factor in $G$ in which every vertex has
positive even degree. A \emph{$[2, 2s]$-factor} of $G$ is a connected even factor of $G$ in which every vertex has  degree at most $2s$. Some results for the
existence of such kind  factors by using forbidden subgraphs
have been appeared, for examples see \cite{Duan}, \cite{LiMingChu}, and \cite{Favaron}.
 Since a hamiltonian cycle is a $[2, 2s]$-factor with $s=1$, the minimum $s$ in a $[2, 2s]$-factor of a graph can be seen as a measure for how close a graph is to
become hamiltonian. Furthermore we know from \cite{Under} that  it is NP-complete to determine whether the square of a graph is hamiltonian. Therefore
the determination of minimum $s$ in a $[2, 2s]$-factor in the square of a graph is also NP-complete.

The result by Fleischner in \cite{FLE} concerning the existence of a hamiltonian cycle (a $[2, 2]$-factor) in the square of 2-connected graph is
well known. Recently, M\"{u}ttel and  Rautenbach in \cite{MUT} gave a shorter proof of this result.

\begin{theorem}\emph{\textbf{\cite{FLE}}}
 \label{Fleischner}
  If $G$ is a 2-connected graph and $v_1$ and $v_2$ are two distinct vertices of $G$, then $G^2$ contains a hamiltonian cycle $C$ such that both edges of $C$
  incident with $v_1$ and one edge of $C$ incident with $v_2$ belong to $G$. Furthermore, if $v_1$ and $v_2$ are neighbors in $C$, then these are three
  distinct edges.
\end{theorem}

Theorem \ref{Fleischner} was a base for proving the following theorem by Abderrezzak et al. in \cite{EL} using forbidden subgraphs.
The graph $S(H)$ is obtained from a graph $H$ by subdividing  each edge of $H$ exactly once.

\begin{theorem}\emph{\textbf{\cite{EL}}}
 \label{Abderrezzak}
  If $G$ is a connected graph such that every induced $S(K_{1,3})$ has at least three edges in a block of degree at most 2, then $G^{2}$ is hamiltonian.
\end{theorem}

Theorem \ref{Abderrezzak} was generalized by Ekstein et al. in \cite{EKS} for $[2, 2s]$-factors.

\begin{theorem}\emph{\textbf{\cite{EKS}}}
 \label{Ekstein}
  Let $s$ be a positive integer and $G$ be a connected graph such that every induced $S(K_{1, 2s+1})$ has at least three edges in a block of degree
  at most two. Then $G^2$ has a $[2, 2s]$-factor.
\end{theorem}

Let $G$ be a connected graph. Recall that a graph $G$ is \emph{essentially $k$-edge connected} if deleting less than $k$ edges from $G$ cannot result
in two nontrivial components. In this paper, we shall answer the question how it is for the existence of a $[2, 2s]$-factor in the square of a graph
with 2-edge (or essentially 2-edge)-connectivity instead of (vertex) connectivity of a graph.

A vertex of degree 1 is called {\sl a leaf}. A cut vertex $y$ is {\sl trivial} in $G$, if $y$ is not a cut vertex in $G-M$,
where $M$ is a set of all leaves adjacent  to $y$, otherwise is {\sl non-trivial}. If $M=\{x\}$ and the neighbor of $x$ is a trivial cut vertex of $G$,
then $x$ is called {\sl a bad leaf}. {\sl A trivial bridge} is a cut-edge of $G$ containing a leaf, otherwise is {\sl non-trivial}.
{\sl A bad bridge} is a trivial bridge of $G$ adjacent to a bad leaf. For illustration see Fig.~1.

\begin{figure}[ht]
 \begin{center}
  \label{Figure1}
  $$\beginpicture
   \setcoordinatesystem units <1mm,1mm>
   \setplotarea x from 0 to 0, y from 0 to 30
   \put{$\bullet$} at -30  0
   \put{$x$} at -30 3
   \put{$\bullet$} at -20  0
   \put{$c_{1}$} at -22 3
   \plot -30 0 -20 0 /
   \put{$b_{1}$} at -25 -3
   \circulararc 360 degrees from -20 0 center at -10 0
   \put{$B_{1}$} at -10 0
   \put{$\bullet$} at -10 -10
   \put{$c_{2}$} at -10 -7
   \put{$\bullet$} at -15 -20
   \put{$y_{1}$} at -15 -23
   \put{$\bullet$} at  -5 -20
   \put{$y_{2}$} at -5 -23
   \plot -15 -20 -10 -10 -5 -20 /
   \put{$b_{2}$} at -16 -15
   \put{$b_{3}$} at  -4 -15
   \put{$\bullet$} at  0  0
   \put{$c_{3}$} at 3 -3
   \put{$\bullet$} at  0 10
   \put{$z$} at 0 13
   \put{$\bullet$} at 15  0
   \plot 15 0 0 0 0 10 /
   \put{$b_{4}$} at 3 6
   \put{$b_{5}$} at 8.5 3
   \circulararc 360 degrees from 15 0 center at 25 0
   \put{$B_{2}$} at 25 0
   \put{$\bullet$} at 25 10
   \put{$c_{4}$} at 25 13
   \circulararc 360 degrees from 25 10 center at 25 20
   \put{$B_{3}$} at 25 20
  \endpicture$$
  \caption{In this graph, $c_{1}, c_{2}$ are trivial cut vertices, $c_{3}, c_{4}$ are non-trivial cut vertices, $x$ is a bad leaf, $y_{1}, y_{2}, z$ are
  leaves, $b_{1}$ is a bad bridge, $b_{2}, b_{3}, b_{4}$ are trivial bridges, $b_{5}$ is a non-trivial bridge, and $B_{1}, B_{2}, B_{3}$ are cyclic blocks.}
 \end{center}
\end{figure}

\begin{figure}[ht]
  \begin{center}
  $$\beginpicture
    \setcoordinatesystem units <1mm,1mm>
    \setplotarea x from 0 to 0, y from 0 to 30
    \put{$\bullet$} at -20  0
    \put{$\bullet$} at  20  0
    \circulararc 360 degrees from -20 0 center at -25 0
    \put{$G_{1}$} at -25 0
    \circulararc 360 degrees from 20 0 center at 25 0
    \put{$G_{2}$} at 25 0
    \put{$\bullet$} at 0 -15
    \put{$v_{1}$} at   3 -15
    \put{$\bullet$} at 0 -10
    \put{$\bullet$} at 0  -5
    \put{$v_{2}$} at   3  -5
    \put{$\bullet$} at 0   0
    \plot 0 -15 0 -10 -20 0 0 0 0 -5 0 0 20 0 0 -10 /
    \put{$\vdots$} at 0  6.25
    \put{$\bullet$} at 0  10 \put{$x$} at -19.2 3 \put{$y$} at 19.2 3
    \put{$\bullet$} at -25 5
    \put{$\bullet$} at 25 5
    \circulararc -180 degrees from -25  5 center at 0 5
    \put{$v_{4s+1}$} at 6 15
    \put{$\bullet$} at 0  15
    \plot -20 0 0 10 0 15 0 10 20 0 /
   \endpicture$$\label{Figure2}
   \caption{Essentially 2-edge connected graphs $G$ such that their square contains no $[2,2s]$-factor, where $G_{1}$ and $G_{2}$ are
            any essentially 2-edge connected graphs.}
  \end{center}
 \end{figure}

 Firstly, we look at the graph in Fig.~2, from which one may see the following result.
\begin{theorem}
 \label{Main-}
 For any fixed positive integer $s$, there exists an infinite class of essentially 2-edge-connected graphs $G$ such that $G^{2}$ has no $[2, 2s]$-factor, even
 if the resulting graph obtained from $G$ by deleting its all leaves is 2-connected.
\end{theorem}

\begin{proof}
 Note that the graph $G$ in Fig.~2 is an essentially 2-edge-connected graph. Since every leaf $v_i$ of $G$ has degree exactly 3 in $G^{2}$, at least one
 edge of $v_ix, v_iy$ have to be used in any possible $[2,4]$-factor of $G^2$. Therefore, $G^2$ has no $[2, 2s]$-factor since $G$ has  $4s+1$ such leaves.
\end{proof}

On the other hand, we may show the following result, which is the main result of this paper.

\begin{theorem}
 \label{Main++}
  Let $G$ be a connected graph without non-trivial bridges and without any two bad leaves at distance exactly 4. Then $G^{2}$ has a $[2,4]$-factor.
\end{theorem}

The following corollaries are immediate consequences of Theorem~\ref{Main++}.

\begin{corollary}
 \label{Cor1}
  If $G$ is a 2-edge connected graph, then $G^2$ contains a $[2,4]$-factor.
\end{corollary}

\begin{corollary}
 \label{Cor2}
  If $G$ is an essentially 2-edge connected graph without bad leaves, then $G^2$ contains a $[2,4]$-factor.
\end{corollary}

\begin{corollary}
 \label{Main+}
 Let $G$ be a connected graph without non-trivial bridges. If any two bad leaves have distance at least 5 in $G$, then $G^{2}$ has a $[2,4]$-factor.
\end{corollary}

 Note that the graph in Fig.~2 also shows that the distance 5 in Corollary~\ref{Main+} can not be replaced by distance 4.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A \emph{Useful lemma}}

Before presenting this lemma, we need some additional notation. {\sl Block graph} of a graph $G$, denoted by $BC(G)$, is the graph whose vertex set
consists of all blocks and cut vertices of $G$, and two vertices are adjacent in $BC(G)$ if one of them is a block of $G$ and the second one is its vertex.
It is easy to see that $BC(G)$ is a tree for a connected graph $G$. Note that for any tree, we may choose any vertex as its root. Hence without loss of
generality, we may assume that $B_1, \ldots, B_t$ be all blocks of $G$ such that $B_1$ corresponds to the root of $BC(G)$. For a cut-vertex $v$ of $G$,
{\sl the parent block} of $v$ is the block containing $v$ and its corresponding vertex  in $BC(G)$ has the smallest distance to the root of $BC(G)$.
The remaining blocks containing $v$ are called  {\sl children blocks} of $v$ with respect to the root of $BC(G)$.

The following lemma, we call it a \emph{Useful lemma}, is a key for the proof of  our main result (Theorem ~\ref{Main++}).

\begin{lemma}(Useful lemma)
 \label{Main}
  Let $G$ be a connected graph without non-trivial bridges and without bad leaves (except $K_{1,2}, K_{1,3}$) and $u$ be a vertex of $G$ that is neither
  a cut vertex nor a leaf (if any).

  \noindent
  Then $G^{2}$ has a $[2,4]$-factor $F$ such that
  \begin{itemize}
   \item [a)] $d_F(x)=2$ for any vertex $x$ that is not a cut vertex of $G$;
   \item [b)] both edges of $F$ incident with $u$ belong to $G$;
   \item [c)] for each cut vertex $y$ of $G$ it holds that $d_F(y)=4$ and at least two edges of $F$ incident with $y$ belong to $G$,
              moreover if $y$ is a trivial cut vertex, then these two edges are trivial bridges;
   \item [d)] for any cut vertex $y$ of $G$, the two edges incident with $u$ in $F$ are distinct from the two edges incident with $y$ in $F$
              as specified in (c);
   \item [e)] for any two cut vertices $y_1$ and $y_2$ of $G$, the two edges of $F$ incident with $y_1$ as specified in $(c$) are distince from those
              with $y_2$ as specified in~(c).
  \end{itemize}
\end{lemma}

\begin{proof}
 If $G$ is $K_{1,s}$, for $s\geq4$, then $G^2$ is a complete graph and the result is obvious. Now we assume that $G$ contains at least one cyclic block and $G'=G-M$,
 where $M$ is a set of all leaves adjacent with all trivial cut vertices of $G$.

 Let $\mathbb{O}=B_{1}, B_{2}, \dots, B_{k}$ be an ordering of all blocks of $G'$ such that either $u\in V(B_{1})$, if any, or we choose arbitrary cyclic block
 as $B_{1}$, satisfying the following properties:
\begin{itemize}
\item for any cut vertex $v$ of $G'$, all children blocks of $v$ with respect to the root $r$ of $BC(G')$ corresponding to $B_{1}$ appear consecutively
   in $\mathbb{O}$ such that bridges containing $v$ are in $\mathbb{O}$ before cyclic blocks containing $v$;
\item $\mbox{dist}_{BC(G')}(r, v_{i})<\mbox{dist}_{BC(G')}(r, v_{j})$ implies $i<j$, where $v_{i}, v_{j}$ are vertices of $BC(G')$ corresponding
   to $B_{i}, B_{j}$, respectively.
\end{itemize}
 \medskip

 Then $G'$ is a connected graph without non-trivial bridges and without bad leaves and we prove by induction on $k$ that $(G')^{2}$ contains
 a $[2,4]$-factor $F'$ such that

 \begin{itemize}
  \item[1)] $d_{F'}(x)=2$ for any vertex $x$ that is not a cut vertex of $G$;

  \item[2)] both edges of $F'$ incident with $u$, if any, belong to $B_{1}$;

  \item[3)] for each cut-vertex $y$ of $G'$, it holds that $d_{F'}(y)=4$ and at least two edges of $F'$ incident with $y$ belong to $G'$. Moreover,

   \begin{itemize}
    \item  if $y$ belongs to exactly two blocks of $G'$, then at least two edges of $F'$ incident with $y$ are edges from  the children block of $y$
             with respect to $r$ (the root of $BC(G')$ corresponding to $B_{1}$);

    \item  if $y$ belongs to more than two blocks of $G'$, then at least two edges of $F'$ incident with $y$ are edges from two different children blocks
             of $y$ with respect to $r$.
   \end{itemize}
 \end{itemize}

 For $k=1$, $G'=B_{1}$ and $(G')^{2}$ even has a hamiltonian cycle $C$ such that both edges of $F'$ incident with $u$, if any, belong to $B_{1}$
 by Theorem~\ref{Fleischner}.

 Let $k>1$ and assume that Lemma~\ref{Main} is true for all integers less than $k$. By the definition of $G'$ and $\mathbb{O}$, $B_{k}$ is an end cyclic block of $G'$
 and let $v_{0}$ be the cut vertex of $G'$ with $v_{0}\in V(B_{k})$.

 If $B_{k-1}=v_{0}l$ (i.e. $B_{k-1}$ is a bridge) and $B_{k-1}, B_{k}$ are only children blocks of $v_{0}$ with respect to $r$, then
 we set $G_{1}=G' - \{V(B_{k})\cup\{l\}\setminus\{v_{0}\}\}$, otherwise we set $G_{2}=G' - \{V(B_{k})\setminus\{v_{0}\}\}$. Hence $G_{1}, G_{2}$ are
 connected graphs without non-trivial bridges and without bad leaves and have $k-2, k-1$ blocks, respectively. Hence by the induction hypothesis,
 $(G_{1})^{2}, (G_{2})^{2}$ have a $[2,4]$-factor $F_{1}, F_{2}$ with properties 1), 2), and~3), respectively.

 By Theorem~\ref{Fleischner}, there is a Hamiltonian cycle $C$ in $(B_{k})^{2}$ such that two edges $f_{1}, f_{2}$ of $C$ incident with $v_{0}$ belong
 to $B_{k}$ and thus belong to $G'$.

 \medskip

 \noindent
 \emph{Case 1:} $G_{1}$ exists.

  Let $f_{1}=v_{0}v_{k}$. Then $F'=((F_{1}\cup C)\cup \{v_{0}l,v_{k}l\})\setminus \{f_{1}\}$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).

 \medskip

 \noindent
 \emph{Case 2:} $G_{1}$ does not exist and $v_{0}$ is not a cut vertex in $G_{2}$.

  Hence $v_{0}$ belongs to exactly two blocks of $G'$ and $F'=F_{2}\cup C$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).

 \medskip

 \noindent
 \emph{Case 3:} $G_{1}$ does not exist and $v_{0}$ is a cut vertex in $G_{2}$.

  Let $f_{1}=v_{0}v_{k}$. We consider two possibilities depending on the property~3).

  If exactly two blocks of $G_{2}$ contain $v_{0}$, then by the induction hypothesis $d_{G_{2}}(v_{0})=4$ and there are two edges of $F_{2}$ incident
  with $v_{0}$ from a children block $B_{k-1}$ of $v_{0}$. (Note that $B_{k-1}$ is a cyclic block, since $G_{1}$ does not exist.) Let $e_{k-1}=v_{0}v_{k-1}$
  be  such an edge of $F_{2}$. Since $\mbox{dist}_{G'}(v_{k-1},v_{k})=2$, the edge $v_{k-1}v_{k}$ is an edge of $(G_{2})^{2}$.
  Thus $F'=((F_{2} \cup C)\cup \{v_{k-1}v_{k}\})\setminus \{e_{k-1},f_{1}\}$ is the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).

  If there are more than two blocks of $G_{2}$ containing $v_{0}$, then by the~induction hypothesis $d_{G_{2}}(v_{0})=4$ and there are two edges
  $e_{k-2},e_{k-1}$ of $F_{2}$ incident with $v_{0}$ in $B_{k-2},B_{k-1}$, respectively. Note that it could be $B_{k-2}=e_{k-2}$ or  $B_{k-1}=e_{k-1}$. Let $e_{k-2}=v_{0}v_{k-2}$. Since
  $\mbox{dist}_{G'}(v_{k-2},v_{k})=2$, the edge $v_{k-2}v_{k}$ is an edge of $(G_{2})^{2}$. Thus $F'=((F_{2}\cup C)\cup \{v_{k-2}v_{k}\})\setminus \{e_{k-2},f_{1}\}$ is
  the $[2,4]$-factor of $(G')^{2}$ with properties 1), 2), and 3).

 \medskip

 Now we extend $F'$ to a $[2,4]$-factor $F$ in $G^{2}$ with required properties. Note that the properties 1), 2), a nd 3) imply the
 properties a)-e) in Lemma~\ref{Main}.

 Let $u_{1},u_{2},\dots,u_{t}$ be all trivial cut vertices of $G$ and $l_{i}^{1},l_{i}^{2},\dots,l_{i}^{s_{i}}$ be all leaves incident with $u_{i}$,
 for $i=1,2,\dots,t$. Note that $s_{i}\geq2$, otherwise we have a bad bridge in $G$, a contradiction. For $i=1,2,\dots,t$, let
 $C_{i}=u_{i}l_{i}^{1}l_{i}^{2}\dots l_{i}^{s_{i}}u_{i}$ be cycles in $G^{2}$ and $C'=\cup_{j=1}^{t}C_{j}$. Since $d_{F'}(u_{i})=2$ and
 $u_{i}l_{i}^{1}, l_{i}^{s_{i}}u_{i}$ are edges from $G$, $F=F'\cup C'$ is the $[2,4]$-factor of $G^{2}$ with properties a)-e).
\end{proof}

Note that clearly the square of $K_{1,2}$, $K_{1,3}$ is hamiltonian but there is no $[2,4]$-factor with a vertex of degree 4 in the square of $K_{1,2}$,
$K_{1,3}$, respectively.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem~\ref{Main++}}
In this section we prove   Theorem~\ref{Main++}.
%The proof of Theorem~\ref{Main-} is made  to be convenient for finding an infinite class of graphs.



%Finally we prove Theorem~\ref{Main++}.

\begin{proof}
 Firstly if $G$ is $K_{1,2}$ or $K_{1,3}$, then clearly $G^{2}$ is even hamiltonian.

 Now let $X$ be a set of all bad leaves of $G$ and $G'=G-X$. For $x_{i}\in X$, we denote $y_{x_{i}}$ or only $y_{i}$ its unique neighbor in $G$.
 By Lemma~\ref{Main}, there is a [2,4]-factor $F'$ of $(G')^2$ with properties a)-e). Note that $d_{F'}(y_{i})=2$ for each $y_{i}$.

 By the definition, any two bad leaves have a distance at least 3. Let $X_{0}\subseteq X$ be the set of all bad leaves that has a bad leaf
 at the distance exactly~3 in $G$. Then, for all $x_{i}\in X_{0}$, corresponding $y_{i}$'s induce a subgraph of $G'$ in which all components (denoted by
 $H_{1},H_{2},\dots,H_{s}$) are complete graphs, otherwise we have in $G$ two bad leaves at distance 4, a contradiction.

 Let $V(H_{i})=\{y_{i,1},y_{i,2},\dots,y_{i,t_{i}}\}$, $t_{i}\geq 2$ for $i=1,2,\dots,s$. Then we set

 $$M_{i}=\bigcup_{j=1}^{t_i-1}\{x_{i,j}y_{i,j+1}, x_{i,j+1}y_{i,j}\}~\bigcup~\{x_{i,1}y_{i,1}, x_{i,t_i}y_{i,t_i}\}.$$


 All bad leaves of $X\setminus X_{0}$ are pairwise at distance at least 5 and we divide them into the following three disjoint classes by the following way
 (see Fig.~\ref{Figure3} for illustration):

  \begin{figure}[ht]
  \begin{center}
   \label{Figure3}
   $$\beginpicture
    \setcoordinatesystem units <1mm,1mm>
    \setplotarea x from -30 to 0, y from -10 to 10
    \put{$1)$} at -65 10
    \put{$2)$} at -25 10
    \put{$3)$} at 25 10
    \circulararc 360 degrees from -39 0 center at -50 0
    \circulararc 360 degrees from -21 0 center at -10 0
    \circulararc 360 degrees from  29 0 center at 40 0
    \put{$y_{x}$} at -47 5
    \put{$\bullet$} at -50 5
    \put{$x$} at -47 15
    \put{$\bullet$} at -50 15
    \put{$z_{x}$} at -42 0
    \put{$\bullet$} at -45 0
    \put{\tiny {in $F' $ and $G'$}} at -53 0
    \put{$y_{x}$} at -7 5
    \put{$\bullet$} at -10 5
    \put{$x$} at -7 15
    \put{$\bullet$} at -10 15
    \put{$z_{x}$} at -2 0
    \put{$\bullet$} at -5 0
    \put{\tiny{no cut vertex}} at 9 0
    \put{$y_{x}$} at 43 5
    \put{$\bullet$} at 40 5
    \put{$x$} at 43 15
    \put{$\bullet$} at 40 15
    \put{$z_{x}$} at 48 0
    \put{$\bullet$} at 45 0
    \put{\tiny{cut vertex}} at 57 0
    \put{$z'_{x}$} at 37 0
    \put{$\bullet$} at 40 0
    \put{$z''_{x}$} at 42 -7
    \put{$\bullet$} at 45 -5
    \put{\tiny{in $G'$}} at 42 -2.5
    \plot -45 0 -50 5 -50 15 /
    \plot -5 0 -10 5 -10 15 /
    \plot 45 0 40 5 40 15 /
    \plot 40 0 45 0 45 -5 /
   \endpicture$$
   \caption{Three cases in an ordering of all bad leaves of $X\setminus X_0$ in $G$.}
  \end{center}
 \end{figure}

 \begin{itemize}
  \item[1)] Let $X_1$ be the set of all vertices $x\in X\setminus X_0$ such that there exists a vertex $z_x$ with $y_xz_x\in E(F')\cap E(G')$;
  \item [2)] Let $X_2$ be the set of all vertices $x\in X\setminus (X_0\cup X_1)$ such that there
exists $z_x$, which is not a cut vertex   of $G'$, with $y_xz_x\in E(G')$ (and $y_xz_x\in E(F')$);
  \item [3)] Let $X_3$ be the set of all vertices $x\in X\setminus (X_0\cup X_1\cup X_2)$ (it means that there exists only a cut vertex $z_x$  of $G'$
             with $y_xz_x\in E(G')$ (and $y_xz_x\in E(F')$).
\end{itemize}

 Note that by Lemma~\ref{Main} we have
  \begin{itemize}
   \item  $d_{F'}(z_x)=2$ \ for $x\in X_2$;
   \item  $d_{F'}(z_x)=4$  and at least two edges incident with $z_x$ (namely $z_xz'_x, z_xz''_x)$  are in $E(E')\cap E(G')$   for $x\in X_3.$
  \end{itemize}
 Now set
 $$E_0=\bigcup _{i=1}^{s}M_{i},~~~ E_{1}=\bigcup_{x\in X_1}\{xy_x, xz_x\},~~~ E'_{1}=\bigcup_{x\in X_1}\{y_xz_x\},$$
 $$E_{2}=\bigcup_{x\in X_2}\{xy_x, xz_x, y_xz_x\},$$
 $$E_{3}=\bigcup_{x\in X_3}\{xy_x, xz_{x}, y_{x}z'_{x}\},~~~
   E'_{3}=\bigcup_{x\in X_3}\{z_{x}z'_{x}\}.$$

 For all $x$, $z_x$'s are different, otherwise if $z_{x}=z_{x'}$, for $x\neq x'$, then $xy_xz_x(=z_{x'})y_{x'}x'$ is
 a path of length 4 in $G$ joining two bad leaves, a contradiction. Similarly, none of $z_{x}$'s is a neighbor of a bad leaf in $G$.

 Possibly, $z_{x_{i_{1}}}z_{x_{i_{2}}}\dots z_{x_{i_{k}}}$ is a path in $F'$ for $\{x_{i_{1}},x_{i_{2}},\dots,x_{i_{k}}\}\subseteq X_3$.
 In order to have different edges in $E_{3}$ and $E'_{3}$ we set $z'_{x_j}=z_{x_{j+1}}$, for $j= i_{1},i_{2},\dots,i_{k-1}$, and
 $z'_{x_{i_{k}}}$ as arbitrary neighbor of $z_{x_{i_{k}}}$ in $F'$ and in $G$ different from $z_{x_{i_{k-1}}}$. Note that by 3) and
 Lemma~\ref{Main} such a vertex exists and could be some $z_{x_j}$, for $j\in \{i_{1},i_{2},\dots,i_{k-2}\}$.

 Hence we conclude that $F=(F'\cup (E_{0}\cup E_{1}\cup E_{2}\cup E_{3}))\setminus (E'_{1}\cup E'_{3})$ is a~[2,4]-factor of $G^2$.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}

Now we can answer the question from the Introduction. By Theorem~\ref{Fleischner} we know that the square of a 2-connected graph has a $[2,2s]$-factor
for $s=1$. In this paper we proved that the square of a 2-edge-connected graph has a $[2,2s]$-factor for $s=2$ (Corollary~\ref{Cor1})
and that the square of a essentially 2-edge-connected graph without bad leaves has a $[2,2s]$-factor also for $s=2$ (Corollary~\ref{Cor2}).
In general, there exist essentially 2-edge-connected graphs whose square have no $[2,2s]$-factor for every $s$.  This example of $G$ even exists under an
additional condition that the graph obtained from $G$ by deleting all leaves is 2-connected (Theorem~\ref{Main-}).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\begin{thebibliography}{99}
 \bibitem{Duan} F. Duan, W. Zhang, and G. Wang, \newblock Connected even factors in $\{K_{1,\ell}, K_{1,\ell} + e\}$-free graphs, \newblock {\em Ars Combinatoria}, 115 (2014), 385-389.

 \bibitem{EKS} J. Ekstein, \newblock Hamiltonian cycles in the square of a graph, \newblock {\em The Electronic Journal of Combinatorics} 18 (2011), $\#$P203.

 \bibitem{Holub-Xiong} J. Ekstein, P. Holub, T. Kaiser, L. Xiong, and S. Zhang, \newblock Star subdivisions and connected even factors in the square of a graph, \newblock { \em Discrete Mathematics} 312 (2012), 2574-2578.

 \bibitem{EL} M. El Kadi Abderrezzak, E. Flandrin, and Z. Ryj\'{a}\v{c}ek, \newblock Induced $S(K_{1,3})$ and hamiltonian cycles in the square of a graph, \newblock {\em Discrete Mathematics} 207 (1999), 263-269.

 \bibitem{FAU} R. J. Faudree and R. H. Schelp, \newblock The square of a block is strongly path connected, \newblock { \em Journal of Combinatorial Theory}, Series B 20 (1976), 47-61.



 \bibitem{FLE} H. Fleischner, \newblock In the square of graphs, Hamiltonicity and pancyclicity, hamiltonian connectedness and panconnectedness are equivalent concepts, \newblock {\em Monatshefte f\"{u}r  Mathematik} 82 (1976), 125-149.

 \bibitem{GOU} R. J. Gould and M. S. Jacobson, \newblock Forbidden Subgraphs and Hamiltonian Properties in the Square of a Connected Graph, \newblock {\em Journal of Graph Theory } 8 (1984), 147-154.

 \bibitem{HEN} G. Hendry and W. Vogler, \newblock The square of a $S(K_{1,3})$-free graph is vertex pancyclic, \newblock { \em Journal of Graph Theory} 9 (1985), 535-537.

 \bibitem{CHA} G. Chartrand, A. M. Hobbs, H. A. Jung, S. F. Kapoor, and C. St. J. A. Nash-Williams, \newblock The square of a block is Hamiltonian connected,
\newblock {\em Journal of Combinatorial Theory}, Series B 16 (1974), 290-292.

 \bibitem{CHI} G. L. Chia, S. Ong, and L. Y. Tan, \newblock  On graphs whose square have strong hamiltonian properties, \newblock {\em Discrete Mathematics} 309 (2009), 4608-4613.

 \bibitem{LiMingChu} M. C. Li, L. Xiong and H. J. Broersma, \newblock Connected even factors in claw-free graphs, \newblock {\em Discrete Mathematics} 308 (2008), 2282-2284.


 \bibitem{MUT} J. M\"{u}ttel and D. Rautenbach, \newblock A short proof of the versatile version of Fleischner's theorem, \newblock {\em  Discrete Mathematics} 313 (2013), 1929-1933.

 \bibitem{Favaron} F. Odile and K. Mekkia, \newblock Even factors of larger size, \newblock {\em Journal Graph Theory} 77 (2014), 58-67.

 \bibitem{Under}Paris Underground, \newblock On Graphs with Hamiltonian squares, \newblock {\em  Discrete Mathematics } 21 (1987), 323.

 \bibitem{WES} D. B. West, \newblock Introduction to Graph Theory, Second ed. PrinticeHall, Upper Saddle River, NJ, 2001.
\end{thebibliography}



\end{document}











