http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2015-11-27T12:21:23+11:00Andre Kundgenakundgen@csusm.eduOpen Journal Systems<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><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>http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p1The Spectrum for $3$-Way $k$-Homogeneous Latin Trades2015-11-27T12:21:21+11:00Trent G. Marbachtrent.marbach@uqconnect.edu.auLijun Jijilijun@suda.edu.cn<p>A $\mu$-way $k$-homogeneous Latin trade was defined by Bagheri Gh, Donovan, Mahmoodian (2012), where the existence of $3$-way $k$-homogeneous Latin trades was specifically investigated. We investigate the existence of a certain class of $\mu$-way $k$-homogeneous Latin trades with an idempotent like property. We present a number of constructions for $\mu$-way $k$-homogeneous Latin trades with this property, and show that these can be used to fill in the spectrum of $3$-way $k$-homogeneous Latin trades for all but $196$ possible exceptions.</p><p> </p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p2Genus of the Cartesian Product of Triangles.2015-11-27T12:21:21+11:00Michal Kotrbčíkkotrbcik@fi.muni.czTomaž Pisanskitomaz.pisanski@upr.siWe investigate the orientable genus of $G_n$, the cartesian product of $n$ triangles, with a particular attention paid to the two smallest unsolved cases $n=4$ and $5$. Using a lifting method we present a general construction of a low-genus embedding of $G_n$ using a low-genus embedding of $G_{n-1}$. Combining this method with a computer search and a careful analysis of face structure we show that $30\le \gamma(G_4) \le 37$ and $133 \le\gamma(G_5) \le 190$. Moreover, our computer search resulted in more than $1300$ non-isomorphic minimum-genus embeddings of $G_3$. We also introduce genus range of a group and (strong) symmetric genus range of a Cayley graph and of a group. The (strong) symmetric genus range of irredundant Cayley graphs of $Z_p^n$ is calculated for all odd primes $p$.2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p3An Exact Turán Result for Tripartite 3-Graphs2015-11-27T12:21:21+11:00Adam Sanittadam.sanitt11@ucl.ac.ukJohn Talbotj.talbot@ucl.ac.uk<p>Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p4The Page-Rényi Parking Process2015-11-27T12:21:21+11:00Lucas Geringerin@cmap.polytechnique.frIn the Page parking (or packing) model on a discrete interval (also known as the <em>discrete Rényi packing</em> problem or the <em>unfriendly seating problem</em>), cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain.<br /><br />We use a probabilistic construction of the Page parking to give a short proof of the (known) fact that the proportion of the interval occupied by cars goes to $1-e^{-2}$, when the length of the interval goes to infinity. We also obtain some new consequences on both finite and infinite parkings.2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p5Bounding Tree-Width via Contraction on the Projective Plane and Torus.2015-11-27T12:21:21+11:00Evan Morganemorgan@psu.eduBogdan Oporowskibogdan@math.lsu.eduIf $X$ is a collection of edges in a graph $G$, let $G/X$ denote the contraction of $X$. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective planar graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three. We prove that every toroidal graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three and four, respectively.2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p6Identifying Codes in Vertex-Transitive Graphs and Strongly Regular Graphs2015-11-27T12:21:21+11:00Sylvain Graviersylvain.gravier@ujf-grenoble.frAline Parreaualine.parreau@univ-lyon1.frSara Rotteysara.rottey@ugent.beLeo Stormels@cage.ugent.beÉlise Vandommee.vandomme@ulg.ac.beWe consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and $2\ln(|V|)+1$ where $V$ is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order $|V|^{\alpha}$ with $\alpha \in \{\frac{1}{4},\frac{1}{3},\frac{2}{5}\}$. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p7A Generalization of Graham’s Conjecture2015-11-27T12:21:22+11:00Huanhuan Guanguan1110h@163.comPingzhi Yuanyuanpz@scnu.edu.cnXiangneng Zengjunevab@163.com<p>Let $G$ be a finite Abelian group of order $|G|=n$, and let $S=g_1\cdot\ldots\cdot g_{n-1}$ be a sequence over $G$ such that all nonempty zero-sum subsequences of $S$ have the same length. In this paper, we completely determine the structure of these sequences.</p><p> </p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p8HOMFLY Polynomials of Torus Links as Generalized Fibonacci Polynomials2015-11-27T12:21:22+11:00Kemal Taşköprükemal.taskopru@bilecik.edu.trİsmet Altıntaşialtintas@sakarya.edu.tr<p>The focus of this paper is to study the HOMFLY polynomial of $(2,n)$-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of $ (2,n) $-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental properties. We also show that the HOMFLY polynomial of $ (2,n) $-torus link can be obtained from its Alexander-Conway polynomial or the classical Fibonacci polynomial. We finally give the matrix representations and prove important identities, which are similar to the Fibonacci identities, for the our generalized Fibonacci polynomial and the HOMFLY polynomial of $ (2,n) $-torus link.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p9Random-Player Maker-Breaker games2015-11-27T12:21:22+11:00Michael Krivelevichkrivelev@post.tau.ac.ilGal Kronenberggalkrone@mail.tau.ac.il<p>In a $(1:b)$ Maker-Breaker game, one of the central questions is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erd<span>ő</span>s conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erd<span>ő</span>s Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player game, played the same as an ordinary (biased) Maker-Breaker game, except that one player plays according to his best strategy and claims one element in each round, while the other plays randomly and claims $b$ (or $m$) elements. In fact, for every (ordinary) Maker-Breaker game, there are two different random-player versions; the $(1:b)$ random-Breaker game and the $(m:1)$ random-Maker game. We analyze the random-player version of several classical Maker-Breaker games such as the Hamilton cycle game, the perfect-matching game and the $k$-vertex-connectivity game (played on the edge set of $K_n$). For each of these games we find or estimate the asymptotic values of the bias (either $b$ or $m$) that allow each player to typically win the game. In fact, we provide the "smart" player with an explicit winning strategy for the corresponding value of the bias.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10Walks, Partitions, and Normal Ordering2015-11-27T12:21:22+11:00Askar Dzhumadil'daevdzhuma@hotmail.comDamir Yeliussizovyeldamir@gmail.com<p>We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the $n$-th Weyl algebra. Under several specifications, we study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the $\lambda$-Stirling numbers.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p11Fractional Coloring of Triangle-Free Planar Graphs2015-11-27T12:21:22+11:00Zdeněk Dvořákrakdver@iuuk.mff.cuni.czJean-Sébastien Serenijean-sebastien.sereni@loria.frJan Volechonza@ucw.cz<p>We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-3/(3n+1)$.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p12On the Real-Rootedness of the Descent Polynomials of $(n-2)$-Stack Sortable Permutations2015-11-27T12:21:22+11:00Philip B. Zhangzhangbiaonk@163.com<p>Bóna conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Brändén proved this conjecture by establishing a more general result. In this paper, we give another proof of Brändén's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p13Wilf-Classification of Mesh Patterns of Short Length2015-11-27T12:21:22+11:00Ísak Hilmarssonisakh08@ru.isIngibjörg Jónsdóttiringibjorg08@ru.isSteinunn Sigurðardóttirsteinunns08@ru.isLína Viðarsdóttirsigridurlv08@ru.isHenning Ulfarssonhenningu@ru.is<p>This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By enumerating some of the remaining classes we bring that upper-bound further down to 56. Finally, we conjecture that the actual number of Wilf-classes of mesh patterns of length 2 is 46.</p>2015-10-16T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p14Thin Edges in Braces2015-11-27T12:21:22+11:00Cláudio L. Lucchesilucchesi@gmail.comMarcelo H. de Carvalhomhc@facom.ufms.brU. S. R. Murtyusrmurty@math.uwaterloo.caThe <em>bicontraction</em> of a vertex $v$ of degree two in a graph, with precisely two neighbours $v_1$ and $v_2$, consists of shrinking the set $\{v_1,v,v_2\}$ to a single vertex. The <em>retract</em> of a matching covered graph $G$, denoted by $\widehat{G}$, is the graph obtained from $G$ by repeatedly bicontracting vertices of degree two. Up to isomorphism, the retract of a matching covered graph $G$ is unique. If $G$ is a brace on six or more vertices, an edge $e$ of $G$ is <em>thin</em> if $\widehat{G-e}$ is a brace. A thin edge $e$ in a simple brace $G$ is <em>strictly thin</em> if $\widehat{G-e}$ is a simple brace. Theorems concerning the existence of strictly thin edges have been used (implicitly by McCuaig (Pólya's Permanent Problem, <em>Electron. J. of Combin.</em>, 11, 2004) and explicitly by the authors (On the Number of Perfect Matchings in a Bipartite Graph, <em>SIAM J. Discrete Math.</em>, <strong>27</strong>, 940-958, 2013)) as inductive tools for establishing properties of braces.<br /><br />Let $G$ and $J$ be two distinct braces, where $G$ is of order six or more and $J$ is a simple matching minor of $G$. It follows from a theorem of McCuaig (Brace Generation, <em>J. Graph Theory</em>, <strong>38</strong>, 124-169, 2001) that $G$ has a thin edge $e$ such that $J$ is a matching minor of $G-e$. In Section 2, we give an alternative, and simpler proof, of this assertion. Our method of proof lends itself to proving stronger results concerning thin edges.<br /><br />Let ${\cal G}^+$ denote the family of braces consisting of all prisms, all Möbius ladders, all biwheels, and all extended biwheels. Strengthening another result of McCuaig on brace generation, we show that every simple brace of order six or more which is not a member of ${\cal G}^+$ has at least two strictly thin edges. We also give examples to show that this result is best possible.<br /><br /><br />2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p15The Maximal Length of a Gap between $r$-Graph Turán Densities2015-11-27T12:21:22+11:00Oleg PikhurkoO.Pikhurko@warwick.ac.uk<p>The Tur<span>á</span>n density $\pi(\cal F)$ of a family $\cal F$ of $r$-graphs is the limit as $n\to\infty$ of the maximum edge density of an $\cal F$-free $r$-graph on $n$ vertices. Erd<span>ő</span>s [Israel J. Math, 2 (1964):183—190] proved that no Tur<span>á</span>n density can lie in the open interval $(0,r!/r^r)$. Here we show that any other open subinterval of $[0,1]$ avoiding Tur<span>á</span>n densities has strictly smaller length. In particular, this implies a conjecture of Grosu [<a href="http://arxiv.org/abs/1403.4653">arXiv:1403.4653</a>, 2014].</p>2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p16Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups2015-11-27T12:21:22+11:00Silvia Goodenoughsilvia.goodenough@lipn.univ-paris13.frChristian Lavaultlavault@lipn.univ-paris13.fr<p>In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg—Weyl, its Bargmann—Fock representation with differential operators and the associated one-parameter group.</p><p>Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e., substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the "striped" Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.</p>2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p17Rank and Crank Analogs for some Colored Partitions2015-11-27T12:21:22+11:00Roberta R. Zhouzhourui@neuq.edu.cnWenlong Zhangwenlongzhang@dlut.edu.cn<p>We establish some rank and crank analogs for partitions into distinct colors and give combinatorial interpretations for colored partitions such as partitions defined by Toh, Zhang and Wang congruences modulo 5, 7.</p>2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p18$h$-Polynomials via Reduced Forms2015-11-27T12:21:22+11:00Karola Meszaroskarolam@gmail.comThe flow polytope $F_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $F_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of $h$-polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi-classical Yang-Baxter algebra.2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p19Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture2015-11-27T12:21:22+11:00Florian Lehnermail@florian-lehner.netRögnvaldur G. Möllerroggi@hi.is<pre><!--StartFragment-->A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing <span>2</span>-colouring.</pre><pre> </pre><pre>We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.<!--EndFragment--></pre>2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p20Reconstructing Permutations from Identification Minors2015-11-27T12:21:22+11:00Erkko Lehtonenerkko.lehtonen@iki.fiWe consider the problem whether a permutation of a finite set is uniquely determined by its identification minors. While there exist non-reconstructible permutations of every set with two, three, or four elements, we show that every permutation of a finite set with at least five elements is reconstructible from its identification minors. Moreover, we provide an algorithm for recovering a permutation from its deck. We also discuss a generalization of this reconstruction problem, as well as the related set-reconstruction problem.2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p21Normally Regular Digraphs2015-11-27T12:21:22+11:00Leif K Jørgensenleif@math.aau.dk<p>A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $\mu$ common out-neighbours, a pair of vertices connected by an edge in one direction have $\lambda$ common out-neighbours and a pair of vertices connected by edges in both directions have $2\lambda-\mu$ common out-neighbours. We often assume that two vertices can not be connected in both directions. <br /><br />We prove that the adjacency matrix of a normally regular digraph is normal. A connected $k$-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than $k$ are on one circle in the complex plane. We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs are Cayley graphs of abelian groups and the construction is then based on a generalization of difference sets.</p><p>We also show connections to other combinatorial objects: strongly regular graphs, symmetric 2-designs and association schemes.<br /><br /></p>2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p22Enumeration of Lozenge Tilings of Halved Hexagons with a Boundary Defect2015-11-27T12:21:22+11:00Ranjan Rohatgirrohatgi@indiana.edu<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>We generalize a special case of a theorem of Proctor on the enumeration of lozenge tilings of a hexagon with a maximal staircase removed using Kuo’s graphical condensation method. Additionally, we prove a formula for a weighted version of the given region. The result also extends work of Ciucu and Fischer. By applying the factorization theorem of Ciucu, we are also able to generalize a special case of MacMahon’s boxed plane partition formula. </span></p></div></div></div>2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p23Words with many Palindrome Pair Factors2015-11-27T12:21:22+11:00Adam Borchertadamdborchert@gmail.comNarad Rampersadnarad.rampersad@gmail.comMotivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.2015-10-30T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p24Clustered Planarity Testing Revisited2015-11-27T12:21:22+11:00Radoslav Fulekradoslav.fulek@gmail.comJan Kynčlkyncl@kam.mff.cuni.czIgor Malinovićigor.malinovic@epfl.chDömötör Pálvölgyidomotorp@gmail.comThe Hanani–Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani–Tutte theorem in the case when each cluster induces a connected subgraph.<br /><br />Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident with at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.2015-11-13T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p25Cataloguing PL 4-Manifolds by Gem-Complexity2015-11-27T12:21:22+11:00Maria Rita Casalicasali@unimore.itPaola Cristoforipaola.cristofori@unimore.it<p>We describe an algorithm to subdivide automatically a given set of PL $n$-manifolds (via <em>coloured triangulations</em> or, equivalently, via <em>crystallizations</em>) into classes whose elements are PL-homeomorphic. The algorithm, implemented in the case <em>n=4</em>, succeeds to solve completely the PL-homeomorphism problem among the catalogue of all closed connected PL 4-manifolds up to gem-complexity 8 (i.e., which admit a coloured triangulation with at most 18 4-simplices).<br /><br />Possible interactions with the (not completely known) relationship among different classification in TOP and DIFF=PL categories are also investigated. As a first consequence of the above PL classification, the non-existence of exotic PL 4-manifolds up to gem-complexity 8 is proved. Further applications of the tool are described, related to possible PL-recognition of different triangulations of the <em>K3</em>-surface.</p>2015-11-13T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p26A Linear Bound towards the Traceability Conjecture2015-11-27T12:21:23+11:00Susan A. van Aardtsusan@cube.co.zaJean E. Dunbarjean.dunbar@converse.eduMarietjie Frickmarietjie.frick@gmail.comNicolas Lichiardopolnicolas.lichiardopol@neuf.frA digraph is <em>k</em>-traceable if its order is at least <em>k</em> and each of its subdigraphs of order <em>k</em> is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so <em>k</em>-traceable oriented graphs may be regarded as generalized tournaments. It is well-known that all tournaments are traceable. We denote by <em>t</em>(<em>k</em>) the smallest integer bigger than or equal to <em>k</em> such that every <em>k</em>-traceable oriented graph of order at least <em>t</em>(<em>k</em>) is traceable. The Traceability Conjecture states that <em>t</em>(<em>k</em>) ≤ <em>2k-1</em> for every <em>k </em>≥ <em>2</em> [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008]. We show that for <em>k </em>≥ 2, every <em>k</em>-traceable oriented graph with independence number 2 and order at least 4<em>k-</em>12 is traceable. This is the last open case in giving an upper bound for <em>t</em>(<em>k</em>) that is linear in <em>k</em>.2015-11-13T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p27On Generalizations of the Petersen Graph and the Coxeter Graph2015-11-27T12:21:23+11:00Marko Orelmarkoorel.math@gmail.com<p>In this note we consider two related infinite families of graphs, which generalize the Petersen and the Coxeter graph. The main result proves that these graphs are cores. It is determined which of these graphs are vertex/edge/arc-transitive or distance-regular. Girths and odd girths are computed. A problem on hamiltonicity is posed.</p>2015-11-13T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p28Inversion Formulae on Permutations Avoiding 3212015-11-27T12:21:23+11:00Pingge Chenchenpingge@hnu.edu.cnZhousheng Mei15616197873@163.comSuijie Wangwangsuijie@hnu.edu.cn<p>We will study the inversion statistic of $321$-avoiding permutations, and obtain that the number of $321$-avoiding permutations on $[n]$ with $m$ inversions is given by<br />\[<br />|\mathcal {S}_{n,m}(321)|=\sum_{b \vdash m}{n-\frac{\Delta(b)}{2}\choose l(b)}.<br />\]<br />where the sum runs over all compositions $b=(b_1,b_2,\ldots,b_k)$ of $m$, i.e.,<br />\[<br />m=b_1+b_2+\cdots+b_k \quad{\rm and}\quad b_i\ge 1,<br />\]<br />$l(b)=k$ is the length of $b$, and $\Delta(b):=|b_1|+|b_2-b_1|+\cdots+|b_k-b_{k-1}|+|b_k|$. We obtain a new bijection from $321$-avoiding permutations to Dyck paths which establishes a relation on inversion number of $321$-avoiding permutations and valley height of Dyck paths.</p>2015-11-13T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p29Exact Forbidden Subposet Results using Chain Decompositions of the Cycle2015-11-27T12:21:23+11:00Abhishek Methukuabhishekmethuku@gmail.comCasey Tompkinsctompkins496@gmail.comWe introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the size of the largest family of subsets of $[n]$ not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities.2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p30Extended Formulation for CSP that is Compact for Instances of Bounded Treewidth2015-11-27T12:21:23+11:00Petr Kolmankolman@kam.mff.cuni.czMartin Kouteckýkoutecky@kam.mff.cuni.czIn this paper we provide an extended formulation for the class of constraint satisfaction problems and prove that its size is polynomial for instances whose constraint graph has bounded treewidth. This implies new upper bounds on extension complexity of several important NP-hard problems on graphs of bounded treewidth.2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p31Digraph Representations of 2-closed Permutation Groups with a Normal Regular Cyclic Subgroup2015-11-27T12:21:23+11:00Jing Xuxujing@cnu.edu.cn<p>In this paper, we classify 2-closed (in Wielandt's sense) permutation groups which contain a normal regular cyclic subgroup and prove that for each such group $G$, there exists a circulant $\Gamma$ such that $\mathrm{Aut} (\Gamma)=G$.</p>2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p32Developments in the Khintchine-Meinardus Probabilistic Method for Asymptotic Enumeration2015-11-27T12:21:23+11:00Boris L. Granovskymar18aa@techunix.technion.ac.ilDudley Starkd.s.stark@qmul.ac.uk<p>A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the Taylor coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of the above generating functions. This allows novel applications of the method. In particular, we prove rigorously the asymptotics of Gentile statistics and derive the asymptotics of combinatorial objects with distinct components.<br /><br /></p>2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p33On the Existence of Certain Optimal Self-Dual Codes with Lengths Between 74 and 1162015-11-27T12:21:23+11:00Tao Zhangzhant220@163.comJerod Michelsamarkand_city@126.comTao Fengtfeng@zju.edu.cnGennian Gegnge@zju.edu.cnThe existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a prime. These results are applied to construct new self-dual codes with length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes, four of which have new weight enumerators. We also show that there are at least $141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to equivalence. Meanwhile, we give some restrictions on the weight enumerators of singly even self-dual codes. We use these restrictions to exclude some possible weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and $100$.2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p34Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles2015-11-27T12:21:23+11:00Louis DeBiasiodebiasld@miamioh.eduTheodore Mollamolla@illinois.eduIn 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p35Applications of Integer Programming Methods to Cages2015-11-27T12:21:23+11:00Frans J.C.T. de RuiterF.J.C.T.deRuiter@tilburguniversity.eduNorman L. BiggsN.L.Biggs@lse.ac.uk<p>The aim of this paper is to construct new small regular graphs with girth $7$ using integer programming techniques. Over the last two decades solvers for integer programs have become more and more powerful and have proven to be a useful aid for many hard combinatorial problems. Despite successes in many related fields, these optimisation tools have so far been absent in the quest for small regular graphs with a given girth. Here we illustrate the power of these solvers as an aid to construct small regular girth $7$ graphs from girth $8$ cages.</p>2015-11-27T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p36Lifespan in a Primitive Boolean Linear Dynamical System2015-11-27T12:21:23+11:00Yaokun Wuykwu@sjtu.edu.cnYinfeng Zhufengzi@sjtu.edu.cn<p>Let $\mathcal F$ be a set of $k$ by $k$ nonnegative matrices such that every "long" product of elements of $\mathcal F$ is positive. Cohen and Sellers (1982) proved that, then, every such product of length $2^k-2$ over $\mathcal F$ must be positive. They suggested to investigate the minimum size of such $\mathcal F$ for which there exists a non-positive product of length $2^k-3$ over $\mathcal F$ and they constructed one example of size $2^k-2$. We construct one of size $k$ and further discuss relevant basic problems in the framework of Boolean linear dynamical systems. We also formulate several primitivity properties for general discrete dynamical systems.<br /><br /><br /></p>2015-11-27T00:00:00+11:00