The Electronic Journal of Combinatorics
http://www.combinatorics.org/ojs/index.php/eljc
<p>The Electronic Journal of Combinatorics (E-JC) is a fully-refereed electronic journal with <em>very high standards,</em> publishing papers of substantial content and interest in all branches of discrete mathematics, including combinatorics, graph theory, and algorithms for combinatorial problems. The journal is <em>completely free</em> for both authors and readers. Authors retain the copyright of their articles. Articles published in E-JC are reviewed on MathSciNet and ZBMath, and are indexed by Web of Science. The latest papers are always available by clicking on the tab marked "Current" near the top of the page. You can also locate papers using the search facility on the right hand side of this page.</p><p>E-JC was founded in 1994 by Herbert S. Wilf and Neil Calkin, making it one of the oldest electronic journals.</p>en-US<p>The copyright of published papers remains with the authors. We only require your agreement that we publish it, as described in the following publication release agreement:</p><ol><li>This is an agreement between the Electronic Journal of Combinatorics (the "Journal"), and the copyright owner (the "Owner") of a work (the "Work") to be published in the Journal.</li><li>The Owner warrants that s/he has the full power and authority to enter into this Agreement and to grant the rights granted in this Agreement.</li><li>The Owner hereby grants to the Journal a worldwide, irrevocable, royalty free license to publish or distribute the Work, to enter into arrangements with others to publish or distribute the Work, and to archive the Work.</li><li>The Owner agrees that further publication of the Work, with the same or substantially the same content as appears in the Journal, will include an acknowledgement of prior publication in the Journal.</li></ol>akundgen@csusm.edu (Andre Kundgen)bdm@cs.anu.edu.au (Brendan McKay)Wed, 01 Jul 2015 14:19:19 +1000OJS 2.3.6.0http://blogs.law.harvard.edu/tech/rss60On Obstacle Numbers
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p1
<p>The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala et al. (2012) show that there exist graphs with $n$ vertices having obstacle number in $\Omega(n/\log n)$. In this note, we up this lower bound to $\Omega(n/(\log\log n)^2)$. Our proof makes use of an upper bound of Mukkamala et al. on the number of graphs having obstacle number at most $h$ in such a way that any subsequent improvements to their upper bound will improve our lower bound.</p>Vida Dujmović, Pat Morinhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p1Wed, 01 Jul 2015 00:00:00 +1000On a Conjecture of Thomassen
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p2
<p>In 1989, Thomassen asked whether there is an integer-valued function $f(k)$ such that every $f(k)$-connected graph admits a spanning, bipartite $k$-connected subgraph. In this paper we take a first, humble approach, showing the conjecture is true up to a $\log n$ factor.</p>Michelle Delcourt, Asaf Ferberhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p2Wed, 01 Jul 2015 00:00:00 +1000Distinct Parts Partitions without Sequences
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p3
<p>Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.</p>Kathrin Bringmann, Karl Mahlburg, Karthik Natarajhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p3Wed, 01 Jul 2015 00:00:00 +1000Homomesy in Products of Two Chains
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p4
Many invertible actions $\tau$ on a set $\mathcal{S}$ of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit the following property which we dub <strong>homomesy</strong>: the average of $f$ over each $\tau$-orbit in $\mathcal{S}$ is the same as the average of $f$ over the whole set $\mathcal{S}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter's action on certain subposets of Young's Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains. <br /><br /><br /><br />James Propp, Tom Robyhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p4Wed, 01 Jul 2015 00:00:00 +1000Small Regular Graphs of Girth 7
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p5
In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of $(q+1,8)$-cages, for $q$ a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain $(q+1)$-regular graphs of girth 7 and order $2q^3+q^2+2q$ for each even prime power $q \ge 4$, and of order $2q^3+2q^2-q+1$ for each odd prime power $q\ge 5$.M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate, J. Salashttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p5Wed, 01 Jul 2015 00:00:00 +1000Completing Partial Proper Colorings using Hall's Condition
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p6
<pre><!--StartFragment-->In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph <span>$G$</span> with list assignment <span>$L$</span> satisfies <em>Hall's condition</em> if for each subgraph <span>$H$</span> of <span>$G$</span>, the inequality <span>$|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ </span>is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called <em>Hall</em> if $(G,L)$ satisfies Hall's condition. A graph $G$ is <em>Hall</em> $m$-c<em>ompletable</em> for some $m \geq \chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \geq 3$? (2) If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-completable? This paper establishes that for every $m \geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\Delta(G)$-completable. </pre>Sarah Holliday, Jennifer Vandenbussche, Erik E Westlundhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p6Wed, 01 Jul 2015 00:00:00 +1000Acyclic Subgraphs of Planar Digraphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p7
An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on $n$ vertices without directed 2-cycles possesses an acyclic set of size at least $3n/5$. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if $g$ is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least $(1 - 3/g)n$.Noah Golowich, David Rolnickhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p7Wed, 01 Jul 2015 00:00:00 +1000Arithmetic Properties of a Restricted Bipartition Function
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p8
<p>A bipartition of $n$ is an ordered pair of partitions $(\lambda,\mu)$ such that the sum of all of the parts equals $n$. In this article, we concentrate on the function $c_5(n)$, which counts the number of bipartitions $(\lambda,\mu)$ of $n$ subject to the restriction that each part of $\mu$ is divisible by $5$. We explicitly establish four Ramanujan type congruences and several infinite families of congruences for $c_5(n)$ modulo $3$.</p>Jian Liu, Andrew Y.Z. Wanghttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p8Wed, 01 Jul 2015 00:00:00 +1000Strong Turán Stability
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p9
We study maximal $K_{r+1}$-free graphs $G$ of almost extremal size<span>—</span>typically, $e(G)=\operatorname{ex}(n,K_{r+1})-O(n)$. We show that any such graph $G$ must have a large amount of `symmetry': in particular, all but very few vertices of $G$ must have twins. (Two vertices $u$ and $v$ are <em>twins</em> if they have the same neighbourhood.) As a corollary, we obtain a new, short proof of a theorem of Simonovits on the structure of extremal $K_{r+1}$-free graphs of chromatic number at least $k$ for all fixed $k \geq r \geq 2$.Mykhaylo Tyomkyn, Andrew J. Uzzellhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p9Wed, 01 Jul 2015 00:00:00 +1000Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p10
<p>It was shown by LeCompte, Martin, and Owens in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a $Q$-polynomial association scheme of class four which is both $Q$-antipodal and $Q$-bipartite. We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain an upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the best general upper bound for the mutually unbiased Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.</p>Hadi Kharaghani, Sara Sasani, Sho Sudahttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p10Wed, 01 Jul 2015 00:00:00 +1000New Lower Bounds for 28 Classical Ramsey Numbers
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p11
<p>We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$.</p>Geoffrey Exoo, Milos Tatarevichttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p11Fri, 17 Jul 2015 00:00:00 +1000Sign Conjugacy Classes of the Symmetric Groups
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p12
A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.Lucia Morottihttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p12Fri, 17 Jul 2015 00:00:00 +1000Forbidden Triples Generating a Finite set of 3-Connected Graphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p13
<p>For a graph $G$ and a set $\mathcal{F}$ of connected graphs, $G$ is said be $\mathcal{F}$-free if $G$ does not contain any member of $\mathcal{F}$ as an induced subgraph. We let $\mathcal{G} _{3}(\mathcal{F})$ denote the set of all $3$-connected $\mathcal{F}$-free graphs. This paper is concerned with sets $\mathcal{F}$ of connected graphs such that $|\mathcal{F}|=3$ and $\mathcal{G} _{3}(\mathcal{F})$ is finite. Among other results, we show that for an integer $m\geq 3$ and a connected graph $T$ of order greater than or equal to $4$, $\mathcal{G} _{3}(\{K_{4},K_{2,m},T\})$ is finite if and only if $T$ is a path of order $4$ or $5$.</p>Yoshimi Egawa, Jun Fujisawa, Michitaka Furuya, Michael D Plummer, Akira Saitohttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p13Fri, 17 Jul 2015 00:00:00 +1000Generalized Small Schröder Numbers
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p14
<p style="margin: 0px; text-indent: 0px; -qt-block-indent: 0;">We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from $(0,0)$ to $(2n,0)$ with the step set of $\{(k,k), (l,-l), (2r,0)\, |\, k,l,r \in {\bf P}\}$, where ${\bf P}$ is the set of positive integers, which never goes below the $x$-axis, and with no horizontal steps at level 0. We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to $\sum_{k=1}^{n} N(n,k)5^{n-k}$ for $n\geq 1$.</p>JiSun Huh, SeungKyung Parkhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p14Fri, 31 Jul 2015 00:00:00 +1000Rectangular Symmetries for Coefficients of Symmetric Functions
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p15
<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span style="font-size: 8.000000pt; font-family: 'CMR8';">We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles. </span></p></div></div></div>Emmanuel Briand, Rosa Orellana, Mercedes Rosashttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p15Fri, 31 Jul 2015 00:00:00 +1000On the Density of Certain Languages with $p^2$ Letters
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p16
The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\ldots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$ with four different letters. It is also the cardinality of the quotient of $(\mathbb Z_2\times \mathbb Z_2)^n$ under the left action of the special linear group $\mathrm{SL}(2,\mathbb Z)$. In this paper we show how these two interpretations of $x_n$ are related to each other. More generally, for prime numbers $p$ we show a correspondence between a quotient of $(\mathbb Z_p\times\mathbb Z_p)^n$ and a language with $p^2$ letters and words of length $n$.Carlos Segovia, Monika Winklmeierhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p16Fri, 31 Jul 2015 00:00:00 +1000Overpartitions with Restricted Odd Differences
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p17
We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.Kathrin Bringmann, Jehanne Dousse, Jeremy Lovejoy, Karl Mahlburghttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p17Fri, 31 Jul 2015 00:00:00 +1000Local Fusion Graphs and Sporadic Simple Groups
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p18
For a group $G$ with $G$-conjugacy class of involutions $X$, the local fusion graph $\mathcal{F}(G,X)$ has $X$ as its vertex set, with distinct vertices $x$ and $y$ joined by an edge if, and only if, the product $xy$ has odd order. Here we show that, with only three possible exceptions, for all pairs $(G,X)$ with $G$ a sporadic simple group or the automorphism group of a sporadic simple group, $\mathcal{F}(G,X)$ has diameter $2$.John Ballantyne, Peter Rowleyhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p18Fri, 31 Jul 2015 00:00:00 +1000A Relationship Between Generalized Davenport-Schinzel Sequences and Interval Chains
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p19
Let an $(r, s)$-formation be a concatenation of $s$ permutations of $r$ distinct letters, and let a block of a sequence be a subsequence of consecutive distinct letters. A $k$-chain on $[1, m]$ is a sequence of $k$ consecutive, disjoint, nonempty intervals of the form $[a_{0}, a_{1}] [a_{1}+1, a_{2}] \ldots [a_{k-1}+1, a_{k}]$ for integers $1 \leq a_{0} \leq a_{1} < \ldots < a_{k} \leq m$, and an $s$-tuple is a set of $s$ distinct integers. An $s$-tuple stabs an interval chain if each element of the $s$-tuple is in a different interval of the chain. Alon et al. (2008) observed similarities between bounds for interval chains and Davenport-Schinzel sequences, but did not identify the cause.<br /><br />We show for all $r \geq 1$ and $1 \leq s \leq k \leq m$ that the maximum number of distinct letters in any sequence $S$ on $m+1$ blocks avoiding every $(r, s+1)$-formation such that every letter in $S$ occurs at least $k+1$ times is the same as the maximum size of a collection $X$ of (not necessarily distinct) $k$-chains on $[1, m]$ so that there do not exist $r$ elements of $X$ all stabbed by the same $s$-tuple.<br /><br />Let $D_{s,k}(m)$ be the maximum number of distinct letters in any sequence which can be partitioned into $m$ blocks, has at least $k$ occurrences of every letter, and has no subsequence forming an alternation of length $s$. Nivasch (2010) proved that $D_{5, 2d+1}(m) = \Theta( m \alpha_{d}(m))$ for all fixed $d \geq 2$. We show that $D_{s+1, s}(m) = \binom{m- \lceil \frac{s}{2} \rceil}{\lfloor \frac{s}{2} \rfloor}$ for all $s \geq 2$. We also prove new lower bounds which imply that $D_{5, 6}(m) = \Theta(m \log \log m)$ and $D_{5, 2d+2}(m) = \Theta(m \alpha_{d}(m))$ for all fixed $d \geq 3$. <br /><br />Jesse Genesonhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p19Fri, 14 Aug 2015 00:00:00 +1000Absolute Differences Along Hamiltonian Paths
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p20
We prove that if the vertices of a complete graph are labeled with the elements of an arithmetic progression, then for any given vertex there is a Hamiltonian path starting at this vertex such that the absolute values of the differences of consecutive vertices along the path are pairwise distinct. In another extreme case where the label set has small additive energy, we show that the graph actually possesses a Hamiltonian cycle with the property just mentioned. These results partially solve a conjecture by Z.-W. Sun.Francesco Monopolihttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p20Fri, 14 Aug 2015 00:00:00 +1000Closing the Gap on Path-Kipas Ramsey Numbers
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p21
Given two graphs $G_1$ and $G_2$, the Ramsey number $R(G_1, G_2)$ is the smallest integer $N$ such that, for any graph $G$ of order $N$, either $G_1$ is a subgraph of $G$, or $G_2$ is a subgraph of the complement of $G$. Let $P_n$ denote a path of order $n$ and $\widehat{K}_m$ a kipas of order $m+1$, i.e., the graph obtained from a $P_m$ by adding one new vertex $v$ and edges from $v$ to all vertices of the $P_m$.<br /><pre style="-qt-paragraph-type: empty; -qt-block-indent: 0; text-indent: 0px; margin: 0px;"><br />We close the gap in existing knowledge on exact values of the Ramsey numbers $R(P_n,\widehat{K}_m)$ by determining the exact values for the remaining open cases.</pre>Binlong Li, Yanbo Zhang, Halina Bielak, Hajo Broersma, Premek Holubhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p21Fri, 14 Aug 2015 00:00:00 +1000Nondeterministic Automatic Complexity of Overlap-Free and Almost Square-Free Words
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p22
<p>Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 1/2$. We improve that result by showing that $I(\mathbf t)= 1/2$. We prove that the nondeterministic automatic complexity $A_N(x)$ of a word $x$ of length $n$ is bounded by $b(n):=\lfloor n/2\rfloor + 1$. This enables us to define the complexity deficiency $D(x)=b(n)-A_N(x)$. If $x$ is square-free then $D(x)=0$. If $x$ is almost square-free in the sense of Fraenkel and Simpson, or if $x$ is a overlap-free binary word such as the infinite Thue--Morse word, then $D(x)\le 1$. On the other hand, there is no constant upper bound on $D$ for overlap-free words over a ternary alphabet, nor for cube-free words over a binary alphabet.</p><p>The decision problem whether $D(x)\ge d$ for given $x$, $d$ belongs to $\mathrm{NP}\cap \mathrm{E}$.</p>Kayleigh K. Hyde, Bjørn Kjos-Hanssenhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p22Fri, 14 Aug 2015 00:00:00 +1000Zeros of Jones Polynomials of Graphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p23
<p>In this paper, we introduce the Jones polynomial of a graph $G=(V,E)$ with $k$ components as the following specialization of the Tutte polynomial:<br />$$J_G(t)=(-1)^{|V|-k}t^{|E|-|V|+k}T_G(-t,-t^{-1}).$$<br />We first study its basic properties and determine certain extreme coefficients. Then we prove that $(-\infty, 0]$ is a zero-free interval of Jones polynomials of connected bridgeless graphs while for any small $\epsilon>0$ or large $M>0$, there is a zero of the Jones polynomial of a plane graph in $(0,\epsilon)$, $(1-\epsilon,1)$, $(1,1+\epsilon)$ or $(M,+\infty)$. Let $r(G)$ be the maximum moduli of zeros of $J_G(t)$. By applying Sokal's result on zeros of Potts model partition functions and Lucas's theorem, we prove that<br />\begin{eqnarray*}{q_s-|V|+1\over |E|}\leq r(G)<1+6.907652\Delta_G<br />\end{eqnarray*}<br />for any connected bridgeless and loopless graph $G=(V,E)$ of maximum degree $\Delta_G$ with $q_s$ parallel classes. As a consequence of the upper bound, X.-S. Lin's conjecture holds if the positive checkerboard graph of a connected alternating link has a fixed maximum degree and a sufficiently large number of edges.</p>Fengming Dong, Xian'an Jinhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p23Fri, 14 Aug 2015 00:00:00 +1000On the Largest Component of a Hyperbolic Model of Complex Networks
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p24
We consider a model for complex networks that was introduced by Krioukov et al. In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The $N$ points are distributed according to a <em>quasi-uniform</em> distribution, which is a distorted version of the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. <br /><br /> The model is controlled by two parameters $\alpha$ and $\nu$ where, roughly speaking, $\alpha$ controls the exponent of the power-law and $\nu$ controls the average degree. The present paper focuses on the evolution of the component structure of the random graph. We show that<strong> (a)</strong> for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; <strong>(b)</strong> for $\alpha < 1$ and $\nu$ arbitrary with high probability there is a "giant" component of linear order, and<strong> (c)</strong> when $\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\nu$.Michel Bode, Nikolaos Fountoulakis, Tobias Müllerhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p24Fri, 14 Aug 2015 00:00:00 +1000On Edge-Transitive Graphs of Square-Free Order
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p25
We study the class of edge-transitive graphs of square-free order and valency at most $k$. It is shown that, except for a few special families of graphs, only finitely many members in this class are <em>basic</em> (namely, not a normal multicover of another member). Using this result, we determine the automorphism groups of locally primitive arc-transitive graphs with square-free order.Cai Heng Li, Zai Ping Lu, Gai Xia Wanghttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p25Fri, 14 Aug 2015 00:00:00 +1000Linear Transformations Preserving the Strong $q$-log-convexity of Polynomials
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p26
<p>In this paper, we give a sufficient condition for the linear transformation preserving the strong $q$-log-convexity. As applications, we get some linear transformations (for instance, Morgan-Voyce transformation, binomial transformation, Narayana transformations of two kinds) preserving the strong $q$-log-convexity. In addition, our results not only extend some known results, but also imply the strong $q$-log-convexities of some sequences of polynomials.</p>Bao-Xuan Zhu, Hua Sunhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p26Fri, 28 Aug 2015 00:00:00 +1000Connectivity of some Algebraically Defined Digraphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p27
Let $p$ be a prime, $e$ a positive integer, $q = p^e$, and let $\mathbb{F}_q$ denote the finite field of $q$ elements. Let $f_i\colon\mathbb{F}_q^2\to\mathbb{F}_q$ be arbitrary functions, where $1\le i\le l$, $i$ and $l$ are integers. The digraph $D = D(q;\bf{f})$, where ${\bf f}=f_1,\dotso,f_l)\colon\mathbb{F}_q^2\to\mathbb{F}_q^l$, is defined as follows. The vertex set of $D$ is $\mathbb{F}_q^{l+1}$. There is an arc from a vertex ${\bf x} = (x_1,\dotso,x_{l+1})$ to a vertex ${\bf y} = (y_1,\dotso,y_{l+1})$ if $x_i + y_i = f_{i-1}(x_1,y_1)$ for all $i$, $2\le i \le l+1$. In this paper we study the strong connectivity of $D$ and completely describe its strong components. The digraphs $D$ are directed analogues of some algebraically defined graphs, which have been studied extensively and have many applications.Aleksandr Kodess, Felix Lazebnikhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p27Fri, 28 Aug 2015 00:00:00 +1000Weight of 3-Paths in Sparse Plane Graphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p28
<p>We prove precise upper bounds for the minimum weight of a path on three vertices in several natural classes of plane graphs with minimum degree 2 and girth $g$ from 5 to 7. In particular, we disprove a conjecture by S. Jendrol' and M. Maceková concerning the case $g=5$ and prove the tightness of their upper bound for $g=5$ when no vertex is adjacent to more than one vertex of degree 2. For $g\ge8$, the upper bound recently found by Jendrol' and Maceková is tight.</p>V. A. Aksenov, O. V. Borodin, A. O. Ivanovahttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p28Fri, 28 Aug 2015 00:00:00 +1000On the Game Domination Number of Graphs with Given Minimum Degree
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p29
<p>In the domination game, introduced by Bre<span>š</span>ar, Klavžar, and Rall in 2010, Dominator and Staller alternately select a vertex of a graph $G$. A move is legal if the selected vertex $v$ dominates at least one new vertex <span>– </span>that is, if we have a $u\in N[v]$ for which no vertex from $N[u]$ was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the number of vertices selected. The goal of Dominator is to minimize whilst that of Staller is to maximize the length of the game. The game domination number $\gamma_g(G)$ of $G$ is the length of the domination game in which Dominator starts and both players play optimally. In this paper we establish an upper bound on $\gamma_g(G)$ in terms of the minimum degree $\delta$ and the order $n$ of $G$. Our main result states that for every $\delta \ge 4$,$$\gamma_g(G)\le \frac{15\delta^4-28\delta^3-129\delta^2+354\delta-216}{45\delta^4-195\delta^3+174\delta^2+174\delta-216}\; n.$$</p><p>Particularly, $\gamma_g(G) < 0.5139\; n$ holds for every graph of minimum degree 4, and $\gamma_g(G) < 0.4803\; n$ if the minimum degree is greater than 4. Additionally, we prove that $\gamma_g(G) < 0.5574\; n$ if $\delta=3$.</p>Csilla Bujtáshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p29Fri, 28 Aug 2015 00:00:00 +1000The Covering Problem in Rosenbloom-Tsfasman Spaces
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p30
<p>We investigate the covering problem in RT spaces induced by the Rosenbloom-Tsfasman metric, extending the classical covering problem in Hamming spaces. Some connections between coverings in RT spaces and coverings in Hamming spaces are derived. Several lower and upper bounds are established for the smallest cardinality of a covering code in an RT space, generalizing results by Carnielli, Chen and Honkala, Brualdi et al., Yildiz et al. A new construction of MDS codes in RT spaces is obtained. Upper bounds are given on the basis of MDS codes, generalizing well-known results due to Stanton et al., Blokhuis and Lam, and Carnielli. Tables of lower and upper bounds are presented too.</p>André G. Castoldi, Emerson L. Monte Carmelohttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p30Fri, 28 Aug 2015 00:00:00 +1000Covering Partial Cubes with Zones
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p31
<p>A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called <em>zones</em>. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the following:</p><ul><li>cover the cells of a line arrangement with a minimum number of lines,</li><li>select a smallest subset of edges in a graph such that for every acyclic orientation, there exists a selected edge that can be flipped without creating a cycle,</li><li>find a smallest set of incomparable pairs of elements in a poset such that in every linear extension, at least one such pair is consecutive,</li><li>find a minimum-size fibre in a bipartite poset.</li></ul><p>We give upper and lower bounds on the worst-case minimum size of a covering by zones in several of those cases. We also consider the computational complexity of those problems, and establish some hardness results.</p>Jean Cardinal, Stefan Felsnerhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p31Fri, 28 Aug 2015 00:00:00 +1000Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p32
We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.<br /><br />Joy Morris, Pablo Spiga, Gabriel Verrethttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p32Fri, 11 Sep 2015 00:00:00 +1000Generalized Line Graphs: Cartesian Products and Complexity of Recognition
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p33
<p>Putting the concept of line graph in a more general setting, for a positive integer $k$, the $k$-line graph $L_k(G)$ of a graph $G$ has the $K_k$-subgraphs of $G$ as its vertices, and two vertices of $L_k(G)$ are adjacent if the corresponding copies of $K_k$ in $G$ share $k-1$ vertices. Then, 2-line graph is just the line graph in usual sense, whilst 3-line graph is also known as triangle graph. The $k$-anti-Gallai graph $\triangle_k(G)$ of $G$ is a specified subgraph of $L_k(G)$ in which two vertices are adjacent if the corresponding two $K_k$-subgraphs are contained in a common $K_{k+1}$-subgraph in $G$.</p><p>We give a unified characterization for nontrivial connected graphs $G$ and $F$ such that the Cartesian product $G\Box F$ is a $k$-line graph. In particular for $k=3$, this answers the question of Bagga (2004), yielding the necessary and sufficient condition that $G$ is the line graph of a triangle-free graph and $F$ is a complete graph (or vice versa). We show that for any $k\ge 3$, the $k$-line graph of a connected graph $G$ is isomorphic to the line graph of $G$ if and only if $G=K_{k+2}$. Furthermore, we prove that the recognition problem of $k$-line graphs and that of $k$-anti-Gallai graphs are NP-complete for each $k\ge 3$.</p>Aparna Lakshmanan S., Csilla Bujtás, Zsolt Tuzahttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p33Fri, 11 Sep 2015 00:00:00 +1000Regular Graphs are Antimagic
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p34
An undirected simple graph $G=(V,E)$ is called antimagic if there exists an injective function $f:E\rightarrow\{1,\dots,|E|\}$ such that $\sum_{e\in E(u)} f(e)\neq\sum_{e\in E(v)} f(e)$ for any pair of different nodes $u,v\in V$. In this note we prove — with a slight modification of an argument of Cranston et al. — that $k$-regular graphs are antimagic for $k\ge 2$.Kristóf Bérczi, Attila Bernáth, Máté Vizerhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p34Fri, 11 Sep 2015 00:00:00 +1000Bessenrodt-Stanley Polynomials and the Octahedron Recurrence
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p35
We show that a family of multivariate polynomials recently introduced by Bessenrodt and Stanley can be expressed as solution of the octahedron recurrence with suitable initial data. This leads to generalizations and explicit expressions as path or dimer partition functions.Philippe Di Francescohttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p35Fri, 11 Sep 2015 00:00:00 +1000Locally Oriented Noncrossing Trees
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p36
We define an orientation on the edges of a noncrossing tree induced by the labels: for a noncrossing tree (i.e., the edges do not cross) with vertices $1,2,\ldots,n$ arranged on a circle in this order, all edges are oriented towards the vertex whose label is higher. The main purpose of this paper is to study the distribution of noncrossing trees with respect to the indegree and outdegree sequence determined by this orientation. In particular, an explicit formula for the number of noncrossing trees with given indegree and outdegree sequence is proved and several corollaries are deduced from it.<p style="-qt-paragraph-type: empty; -qt-block-indent: 0; text-indent: 0px; margin: 0px;"> </p><p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">Sources (vertices of indegree $0$) and sinks (vertices of outdegree $0$) play a special role in this context. In particular, it turns out that noncrossing trees with a given number of sources and sinks correspond bijectively to ternary trees with a given number of middle- and right-edges, and an explicit bijection is provided for this fact. Finally, the in- and outdegree distribution of a single vertex is considered and explicit counting formulas are provided again.</p><br /><p> </p>Isaac Owino Okoth, Stephan Wagnerhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p36Fri, 11 Sep 2015 00:00:00 +1000Generalized Stirling Permutations and Forests: Higher-Order Eulerian and Ward Numbers
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p37
We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a Riordan inverse pair, for which we also provide a combinatorial interpretation.J. Fernando Barbero G., Jesús Salas, Eduardo J.S. Villaseñorhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p37Fri, 11 Sep 2015 00:00:00 +1000Vizing's Conjecture for Graphs with Domination Number 3 - a New Proof
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p38
<p>Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this note we use a new, transparent approach to prove Vizing's conjecture for graphs with domination number 3; that is, we prove that for any graph $G$ with $\gamma(G)=3$ and an arbitrary graph $H$, $\gamma(G\Box H) \ge 3\gamma(H)$.</p>Boštjan Brešarhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p38Sun, 20 Sep 2015 00:00:00 +1000Finding Domatic Partitions in Infinite Graphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/5089
We investigate the apparent difficulty of finding domatic partitions in graphs using tools from computability theory. We consider nicely presented (i.e., computable) infinite graphs and show that even if the domatic number is known, there might not be any algorithm for producing a domatic partition of optimal size. However, we prove that smaller domatic partitions can be constructed if we restrict to <em>regular</em> graphs. Additionally, we establish similar results for total domatic partitions.Matthew Jura, Oscar Levin, Tyler Markkanenhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/5089Sun, 20 Sep 2015 00:00:00 +1000Iterative Properties of Birational Rowmotion II: Rectangles and Triangles
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p40
Birational rowmotion — a birational map associated to any finite poset $P$ — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of $P$. Continuing our exploration of this birational rowmotion, we prove that it has order $p+q$ on the $\left( p, q\right) $-rectangle poset (i.e., on the product of a $p$-element chain with a $q$-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets.Darij Grinberg, Tom Robyhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p40Sun, 20 Sep 2015 00:00:00 +1000Partial List Colouring of Certain Graphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p41
The <em>partial list colouring conjecture</em> due to Albertson, Grossman, and Haas (2000) states that for every $s$-choosable graph $G$ and every assignment of lists of size $t$, $1 \leq t \leq s$, to the vertices of $G$ there is an induced subgraph of $G$ on at least $\frac{t|V(G)|}{s}$ vertices which can be properly coloured from these lists. In this paper, we show that the partial list colouring conjecture holds true for certain classes of graphs like claw-free graphs, graphs with chromatic number at least $\frac{|V(G)|-1}{2}$, chordless graphs, and series-parallel graphs.Jeannette Janssen, Rogers Mathew, Deepak Rajendraprasadhttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p41Sun, 20 Sep 2015 00:00:00 +1000A Remark on the Tournament Game
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p42
<pre>We study the Maker-Breaker tournament game played on the edge set of a given graph <span>$G$</span>. Two players, Maker and Breaker, claim unclaimed edges of <span>$G$</span> in turns, while Maker additionally assigns orientations to the edges that she claims. If by the end of the game Maker claims all the edges of a pre-defined goal tournament, she wins the game. Given a tournament <span>$T_k$</span> on <span>$k$</span> vertices, we determine the threshold bias for the <span>$(1:b)$</span> <span>$T_k$</span>-tournament game on <span>$K_n$</span>. We also look at the <span>$(1:1)$</span> <span>$T_k$</span>-tournament game played on the edge set of a random graph <span>$</span>{\mathcal{G}_{n,p}}$ and determine the threshold probability for Maker's win. We compare these games with the clique game and discuss whether a random graph intuition is satisfied.</pre><pre> </pre>Dennis Clemens, Mirjana Mikalačkihttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p42Sun, 20 Sep 2015 00:00:00 +1000Genus Ranges of 4-Regular Rigid Vertex Graphs
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p43
<p>A rigid vertex of a graph is one that has a prescribed cyclic order of its incident edges. We study orientable genus ranges of 4-regular rigid vertex graphs. The (orientable) genus range is a set of genera values over all orientable surfaces into which a graph is embedded cellularly, and the embeddings of rigid vertex graphs are required to preserve the prescribed cyclic order of incident edges at every vertex. The genus ranges of 4-regular rigid vertex graphs are sets of consecutive integers, and we address two questions: which intervals of integers appear as genus ranges of such graphs, and what types of graphs realize a given genus range. For graphs with $2n$ vertices ($n>1$), we prove that all intervals $[a, b]$ for all $a<b \leq n$, and singletons $[h, h]$ for some $h\leq n$, are realized as genus ranges. For graphs with $2n-1$ vertices ($n\geq 1$), we prove that all intervals $[a, b]$ for all $a<b \leq n$ except $[0,n]$, and $[h,h]$ for some $h\leq n$, are realized as genus ranges. We also provide constructions of graphs that realize these ranges.</p>Dorothy Buck, Egor Dolzhenko, Natasa Jonoska, Masahico Saito, Karin Valenciahttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p43Sun, 20 Sep 2015 00:00:00 +1000