http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2015-07-31T13:03:57+10:00Andre 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/v22i3p1On Obstacle Numbers2015-07-31T13:03:56+10:00Vida Dujmovićvida@cs.mcgill.caPat Morinmorin@scs.carleton.ca<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>2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p2On a Conjecture of Thomassen2015-07-31T13:03:57+10:00Michelle Delcourtdelcour2@illinois.eduAsaf Ferberasaf.ferber@yale.edu<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>2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p3Distinct Parts Partitions without Sequences2015-07-31T13:03:57+10:00Kathrin Bringmannkbringma@math.uni-koeln.deKarl Mahlburgkarlmahlburg@gmail.comKarthik Natarajkartnat@gmail.com<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>2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p4Homomesy in Products of Two Chains2015-07-31T13:03:57+10:00James Proppjamespropp@gmail.comTom Robytom.roby@uconn.eduMany 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 />2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p5Small Regular Graphs of Girth 72015-07-31T13:03:57+10:00M. Abreumarien.abreu@unibas.itG. Araujo-Pardogaraujo@matem.unam.mxC. Balbuenam.camino.balbuena@upc.eduD. Labbatedomenico.labbate@unibas.itJ. Salasjulian.salas@urv.catIn 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 $q\ge 5$.2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p6Completing Partial Proper Colorings using Hall's Condition2015-07-31T13:03:57+10:00Sarah Hollidayshollid4@kennesaw.eduJennifer Vandenbusschejvandenb@kennesaw.eduErik E Westlundewestlun@kennesaw.edu<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>2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p7Acyclic Subgraphs of Planar Digraphs2015-07-31T13:03:57+10:00Noah Golowichnoah_g@verizon.netDavid Rolnickdrolnick@mit.eduAn 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 $(1 - 3/g)n$.2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p8Arithmetic Properties of a Restricted Bipartition Function2015-07-31T13:03:57+10:00Jian Liuliujian8210@gmail.comAndrew Y.Z. Wangyzwang@uestc.edu.cn<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>2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p9Strong Turán Stability2015-07-31T13:03:57+10:00Mykhaylo Tyomkynm.tyomkyn@bham.ac.ukAndrew J. Uzzellandrew.uzzell@unl.eduWe 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 $k \geq r \geq 2$.2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p10Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes2015-07-31T13:03:57+10:00Hadi Kharaghanikharaghani@uleth.caSara Sasanisasani@uleth.caSho Sudasuda@auecc.aichi-edu.ac.jp<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>2015-07-01T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p11New Lower Bounds for 28 Classical Ramsey Numbers2015-07-31T13:03:57+10:00Geoffrey Exoogeoffrey.exoo@gmail.comMilos Tatarevicmilos.tatarevic@gmail.com<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>2015-07-17T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p12Sign Conjugacy Classes of the Symmetric Groups2015-07-31T13:03:57+10:00Lucia Morottilucia.morotti@matha.rwth-aachen.deA 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.2015-07-17T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p13Forbidden Triples Generating a Finite set of 3-Connected Graphs2015-07-31T13:03:57+10:00Yoshimi Egawatakatou@rs.kagu.tus.ac.jpJun Fujisawafujisawa@fbc.keio.ac.jpMichitaka Furuyamichitaka.furuya@gmail.comMichael D Plummermichael.d.plummer@vanderbilt.eduAkira Saitoasaito@chs.nihon-u.ac.jp<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>2015-07-17T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p14Generalized Small Schröder Numbers2015-07-31T13:03:57+10:00JiSun Huhhyunyjia@yonsei.ac.krSeungKyung Parksparky@yonsei.ac.kr<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>2015-07-31T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p15Rectangular Symmetries for Coefficients of Symmetric Functions2015-07-31T13:03:57+10:00Emmanuel Briandebriand@us.esRosa Orellanarosa.c.orellana@dartmouth.eduMercedes Rosasmrosas@us.es<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>2015-07-31T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p16On the Density of Certain Languages with $p^2$ Letters2015-07-31T13:03:57+10:00Carlos Segoviacsegovia@mathi.uni-heidelberg.deMonika Winklmeiermwinklme@uniandes.edu.coThe 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 $n$.2015-07-31T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p17Overpartitions with Restricted Odd Differences2015-07-31T13:03:57+10:00Kathrin Bringmannkathrinbringmann@googlemail.comJehanne Doussejehanne.dousse@liafa.univ-paris-diderot.frJeremy Lovejoylovejoy@math.cnrs.frKarl Mahlburgmahlburg@math.lsu.eduWe 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.2015-07-31T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p18Local Fusion Graphs and Sporadic Simple Groups2015-07-31T13:03:57+10:00John Ballantynejohn.ballantyne@manchester.ac.ukPeter Rowleypeter.rowley@manchester.ac.ukFor 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 $2$.2015-07-31T00:00:00+10:00