http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2015-05-22T15:32:59+10:00André Kündgenakundgen@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/v22i2p1A Note on Coloring Vertex-Transitive Graphs2015-04-16T11:08:59+10:00Daniel W. Cranstondcranston@vcu.eduLandon Rabernlandon.rabern@gmail.com<p>We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\}$, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $\Delta \ge 13$ we prove the Borodin<span>–</span>Kostochka conjecture, i.e., $\chi\le\max\{\omega,\Delta-1\}$.</p>2015-04-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p2Equipopularity Classes in the Separable Permutations2015-04-14T01:16:09+10:00Michael Albertmalbert@cs.otago.ac.nzCheyne Hombergercheyne.homberger@gmail.comJay Pantonejay.pantone@gmail.com<p>When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length $n$ patterns in the separable permutations is equal to the number of partitions of $n-1$.</p>2015-04-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p3Roudneff's Conjecture for Lawrence Oriented Matroids2015-04-14T01:16:09+10:00Luis Pedro Montejanolpmontejano@gmail.comJorge Luis Ramírez-Alfonsínjramirez@um2.frJ.-P. Roudneff has conjectured that every arrangement of $n\ge 2d+1\ge 5$ (pseudo) hyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ cells bounded by each hyperplane. In this note, we show the validity of this conjecture for arrangements arising from Lawrence oriented matroids.2015-04-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p4Young Diagrams and Intersection Numbers for Toric Manifolds associated with Weyl Chambers2015-04-14T01:16:09+10:00Hiraku Abehirakuabe@globe.ocn.ne.jp<p>We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold.</p>2015-04-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p5Permutations Destroying Arithmetic Structure2015-04-14T01:16:09+10:00Veselin Jungićvjungic@sfu.caJulian Sahasrabudhejulian.sahasra@gmail.com<pre>Given a linear form <span>$C_1X_1 + \cdots + C_nX_n$</span>, with coefficients in the integers, we characterize exactly the countably infinite <span>abelian</span> groups <span>$G$</span> for which there exists a permutation <span>$f$</span> that maps all solutions <span>$(\alpha_1, \ldots , \alpha_n) \in G^n$</span> (with the <span>$\alpha_i$</span> not all equal) to the equation <span>$C_1X_1 + \cdots + C_nX_n = 0 $</span> to non-solutions. This generalises a result of <span>Hegarty</span> about permutations of an <span>abelian</span> group avoiding arithmetic progressions. We also study the finite version of the problem suggested by <span>Hegarty</span>. We show that the number of permutations of <span>$\mathbb{Z}/p\mathbb{Z}$</span> that map all 4-term arithmetic progressions to non-progressions, is asymptotically <span>$e^{-1}p!$</span>.</pre>2015-04-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p6Flag-Transitive Non-Symmetric 2-Designs with $(r,\lambda)=1$ and Alternating Socle2015-04-16T11:07:05+10:00Shenglin Zhouslzhou@scut.edu.cnYajie Wangwangyajie8786@163.comThis paper deals with flag-transitive non-symmetric 2-designs with $(r,\lambda)=1$. We prove that if $\mathcal D$ is a non-trivial non-symmetric $2$-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G\leq Aut(\mathcal D)$ is flag-transitive with $Soc(G)=A_n$ for $n\geq 5$, then $\mathcal D$ is a $2$-$(6,3,2)$ design, the projective space $PG(3,2)$, or a $2$-$(10,6,5)$ design.2015-04-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p7Power Sum Expansion of Chromatic Quasisymmetric Functions2015-04-21T14:45:47+10:00Christos A. Athanasiadiscaath@math.uoa.gr<p>The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.</p>2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p8Motzkin Paths, Motzkin Polynomials and Recurrence Relations2015-04-21T14:45:53+10:00Roy OsteRoy.Oste@UGent.beJoris Van der JeugtJoris.VanderJeugt@UGent.beWe consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof.2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p9From Edge-Coloring to Strong Edge-Coloring2015-04-21T14:45:59+10:00Valentin Borozanvalentin.borozan@gmail.comGerard Jennhwa Changgjchang@math.ntu.edu.twNathann Cohennathann.cohen@gmail.comShinya Fujitashinya.fujita.ph.d@gmail.comNarayanan Narayanannarayana@gmail.comReza Naserasrreza@lri.frPetru Valicovpetru.valicov@lif.univ-mrs.fr<pre><!--StartFragment-->In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: <span>$k$</span>-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set $S(v)$ of colors used by edges incident to a vertex $v$ does not intersect $S(u)$ on more than $k$ colors when $u$ and $v$ are adjacent. We provide some sharp upper and lower bounds for $\chi'_{k\text{-int}}$ for several classes of graphs. For $l$-degenerate graphs we prove that $\chi'_{k\text{-int}}(G)\leq (l+1)\Delta -l(k-1)-1$. We improve this bound for subcubic graphs by showing that $\chi'_{2\text{-int}}(G)\leq 6$. We show that calculating $\chi'_{k\text{-int}}(K_n)$ for arbitrary values of $k$ and $n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of $n$. Furthermore, for complete bipartite graphs we prove that $\chi'_{k\text{-int}}(K_{n,m}) = \left\lceil \frac{mn}{k}\right\rceil$. Finally, we show that computing $\chi'_{k\text{-int}}(G)$ is NP-complete for every $k\geq 1$.</pre>2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p10A Construction of Small $(q-1)$-Regular Graphs of Girth 82015-04-21T14:46:07+10:00M. Abreumarien.abreu@unibas.itG. Araujo-Pardogaraujo@matem.unam.mxC. Balbuenam.camino.balbuena@upc.eduD. Labbatedomenico.labbate@unibas.it<p>In this note we construct a new infinite family of $(q-1)$-regular graphs of girth 8 and order $2q(q-1)^2$ for all prime powers $q\geq 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.</p>2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p11A Generalization of Tokuyama's Formula to the Hall-Littlewood Polynomials2015-04-21T14:46:14+10:00Vineet Guptavineetg@stanford.eduUma Royuma.roy.us@gmail.comRoger Van Peskirpeski@princeton.eduA theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's parametrization for the Schur polynomials. We generalize Tokuyama's formula to the Hall-Littlewood polynomials by extending Tokuyama's statistics. Our result, in addition to specializing to Tokuyama's result and the aforementioned classical results, also yields connections to the monomial symmetric function and a new deformation of Stanley's formula.2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p12A Generalization of Very Odd Sequences2015-04-21T14:46:20+10:00Cheng Yeaw Kumatkcy@nus.edu.sgKok Bin Wongkbwong@um.edu.my<p>Let $\mathbb N$ be the set of positive integers and $n\in \mathbb N$. Let $\mathbf{a}=(a_0,a_1,\dots, a_{n-1})$ be a sequence of length $n$, with $a_i\in \{0,1\}$. For $0\leq k\leq n-1$, let \[ A_k(\mathbf{a})=\sum_{\substack{0\leq i\leq j\leq n-1\\ j-i=k}} a_ia_j.\] The sequence $\mathbf{a}$ is called a very odd sequence if $A_k(\mathbf{a})$ is odd for all $0\leq k\leq n-1$. In this paper, we study a generalization of very odd sequences and give a characterisation of these sequences.</p>2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p13Sprague-Grundy Values of the $\mathcal{R}$-Wythoff Game2015-04-21T14:46:40+10:00Albert Gualbertfgu@gmail.comWe examine the Sprague-Grundy values of the game of $\mathcal{R}$-Wythoff, a restriction of Wythoff's game introduced by Ho, where each move is either to remove a positive number of tokens from the larger pile or to remove the same number of tokens from both piles. Ho showed that the $P$-positions of $\mathcal{R}$-Wythoff agree with those of Wythoff's game, and found all positions of Sprague-Grundy value $1$. We describe all the positions of Sprague-Grundy value $2$ and $3$, and also conjecture a general form of the positions of Sprague-Grundy value $g$.2015-04-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p14Characterization of $(2m,m)$-Paintable Graphs2015-04-29T14:27:37+10:00Thomas Mahoneythomas.r.mahoney@gmail.comJixian Meng15067061960@163.comXuding Zhuxudingzhu@gmail.com<p>In this paper, we prove that for any graph $G$ and any positive integer $m$, $G$ is $(2m,m)$-paintable if and only if $G$ is 2-paintable. It was asked by Zhu in 2009 whether $k$-paintable graphs are $(km,m)$-paintable for any positive integer $m$. Our result answers this question in the affirmative for $k=2$.<br /><br /></p>2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p15Locating-Dominating Sets and Identifying Codes in Graphs of Girth at least 52015-04-29T14:28:03+10:00Camino Balbuenam.camino.balbuena@upc.eduFlorent Foucaudflorent.foucaud@gmail.comAdriana Hansbergahansberg@im.unam.mx<p>Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meeting these bounds.</p>2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p16Permutations Generated by a Depth 2 Stack and an Infinite Stack in Series are Algebraic2015-04-29T14:28:13+10:00Murray Eldermurray.elder@newcastle.edu.auGeoffrey Leegeoffrey.a.lee@uon.edu.auAndrew Rechnitzerandrewr@math.ubc.ca<p>We prove that the class of permutations generated by passing an ordered sequence $12\dots n$ through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length $n$ is encoded by a string of length $3n$. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free grammar to compute the generating function:<br />\[<br />\sum_{n\geq 0} c_n t^n = <br />\frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q}<br />\]<br />where $c_n$ is the number of permutations of length $n$ that can be generated, and $q \equiv q(t) = \frac{1-2t-\sqrt{1-4t}}{2t}$ is a simple variant of the Catalan generating function. This in turn implies that $c_n^{1/n} \to 2+2\sqrt{5}$.</p>2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p17Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices2015-04-29T14:28:19+10:00Robert Davisdavis.robert@uky.edu<p>In Ehrhart theory, the $h^*$-vector of a rational polytope often provides insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal $h^*$-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this paper, we examine the $h^*$-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.</p>2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p18On Modular $k$-Free Sets2015-04-29T14:28:27+10:00Victor Lambertvictor.lambert@math.polytechnique.frLet $n$ and $k$ be integers. A set $A\subset\mathbb{Z}/n\mathbb{Z}$ is $k$-free if for all $x$ in $A$, $kx\notin A$. We determine the maximal cardinality of such a set when $k$ and $n$ are coprime. We also study several particular cases and we propose an efficient algorithm for solving the general case. We finally give the asymptotic behaviour of the minimal size of a $k$-free set in $\left[ 1,n\right]$ which is maximal for inclusion.2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p19Output Sum of Transducers: Limiting Distribution and Periodic Fluctuation2015-04-29T14:28:34+10:00Clemens Heubergerclemens.heuberger@aau.atSara Kropfsara.kropf@aau.atHelmut Prodingerhproding@sun.ac.zaAs a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed. The input of the transducer is a random integer in $[0, N)$. Analogues in higher dimensions are also considered. Sequences defined by a certain class of recursions can be written in this framework.<br /><br />Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established. The periodic fluctuation of the expected value is Hölder continuous and, in many cases, nowhere differentiable. A general formula for the Fourier coefficients of this periodic function is derived. Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers. As an example, the abelian complexity function of the paperfolding sequence is analyzed. This sequence has recently been studied by Madill and Rampersad.2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p20Well-Quasi-Order for Permutation Graphs Omitting a Path and a Clique2015-04-29T14:28:43+10:00Aistis AtminasA.Atminas@warwick.ac.ukRobert Brignallr.brignall@open.ac.ukNicholas KorpelainenN.Korpelainen@derby.ac.ukVadim LozinV.Lozin@warwick.ac.ukVincent Vattervatter@ufl.eduWe consider well-quasi-order for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting $P_5$ and a clique of any size is well-quasi-ordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting $\{P_6,K_6\}$, $\{P_7,K_5\}$, and $\{P_8,K_4\}$ are not well-quasi-ordered.2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p21On the Chromatic Number of the Erdős-Rényi Orthogonal Polarity Graph2015-04-29T14:28:49+10:00Xing Pengx2peng@ucsd.eduMichael Taitmtait@math.ucsd.eduCraig Timmonscraig.timmons@csus.eduFor a prime power $q$, let $ER_q$ denote the Erdős-Rényi orthogonal polarity graph. We prove that if $q$ is an even power of an odd prime, then $\chi ( ER_{q}) \leq 2 \sqrt{q} + O ( \sqrt{q} / \log q)$. This upper bound is best possible up to a constant factor of at most 2. If $q$ is an odd power of an odd prime and satisfies some condition on irreducible polynomials, then we improve the best known upper bound for $\chi(ER_{q})$ substantially. We also show that for sufficiently large $q$, every $ER_q$ contains a subgraph that is not 3-chromatic and has at most 36 vertices.2015-04-29T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p22Some Combinatorial Arrays Related to the Lotka-Volterra System2015-05-14T09:48:05+10:00David Callancallan@stat.wisc.eduShi-Mei Mashimeimapapers@163.comToufik Mansourtmansour@univ.haifa.ac.ilThe purpose of this paper is to investigate several context-free grammars suggested by the Lotka-Volterra system. Some combinatorial arrays, involving the Stirling numbers of the second kind and Eulerian numbers, are generated by these context-free grammars. In particular, we present grammatical characterization of some statistics on cyclically ordered partitions.2015-05-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p23Biembeddings of 2-Rotational Steiner Triple Systems2015-05-14T09:48:13+10:00M. J. Grannellm.j.grannell@open.ac.ukJ. Z. Schroederjzschroeder@gmail.com<p>It is shown that for $v\equiv 1$ or 3 (mod 6), every pair of Heffter difference sets modulo $v$ gives rise to a biembedding of two 2-rotational Steiner triple systems of order $2v+1$ in a nonorientable surface.</p><p> </p>2015-05-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p24Hyperbinary Expansions and Stern Polynomials2015-05-14T09:48:22+10:00Karl Dilcherdilcher@mathstat.dal.caLarry EricksenLE22@cornell.edu<p>We introduce an infinite class of polynomial sequences $a_t(n;z)$ with integer parameter $t\geq 1$, which reduce to the well-known Stern (diatomic) sequence when $z=1$ and are $(0,1)$-polynomials when $t\geq 2$. Using these polynomial sequences, we derive two different characterizations of all hyperbinary expansions of an integer $n\geq 1$. Furthermore, we study the polynomials $a_t(n;z)$ as objects in their own right, obtaining a generating function and some consequences. We also prove results on the structure of these sequences, and determine expressions for the degrees of the polynomials.</p>2015-05-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p25Intersections of Shifted Sets2015-05-14T09:48:31+10:00Mauro Di Nassodinasso@dm.unipi.itWe consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the following: If $A=\{a_n\}$ is such that $a_n=o(n^{k/k-1})$, then the $k$-recurrence set $R_k(A)=\{x\mid |A\cap(A+x)|\ge k\}$ contains the distance sets of arbitrarily large finite sets.2015-05-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p26Schubert Calculus and the Homology of the Peterson Variety2015-05-14T09:48:40+10:00Erik Inskoeinsko@fgcu.edu<p>We use the tight correlation between the geometry of the Peterson variety and the combinatorics the symmetric group to prove that homology of the Peterson variety injects into the homology of the flag variety. Our proof counts the points of intersection between certain Schubert varieties in the full flag variety and the Peterson variety, and shows that these intersections are proper and transverse.</p>2015-05-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p27Cooperative Colorings and Independent Systems of Representatives2015-05-22T15:29:51+10:00Ron Aharoniraharoni@gmail.comRon Holzmanholzman@tx.technion.ac.ilDavid Howardhoward@tx.technion.ac.ilPhilipp Sprüsselpspruessel@math.haifa.ac.il<p>We study a generalization of the notion of coloring of graphs, similar in spirit to that of list colorings: a <em>cooperative coloring</em> of a family of graphs $G_1,G_2, \ldots,G_k$ on the same vertex set $V$ is a choice of independent sets $A_i$ in $G_i$ ($1 \le i \le k)$ such that $\bigcup_{i=1}^kA_i=V$. This notion is linked (with translation in both directions) to the notion of ISRs, which are choice functions on given sets, whose range belongs to some simplicial complex. When the complex is that of the independent sets in a graph $G$, an <em>ISR</em> for a partition of the vertex set of a graph $G$ into sets $V_1,\ldots, V_n$ is a choice of a vertex $v_i \in V_i$ for each $i$ such that $\{v_1,\ldots,v_n\}$ is independent in $G$. Using topological tools, we study degree conditions for the existence of cooperative colorings and of ISRs. A sample result: Three cycles on the same vertex set have a cooperative coloring.</p>2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p28On Embeddings of Circulant Graphs2015-05-22T15:30:03+10:00Marston Conderm.conder@auckland.ac.nzRicardo Grandergrande001@ikasle.ehu.es<p>A circulant of order $n$ is a Cayley graph for the cyclic group $\mathbb{Z}_n$, and as such, admits a transitive action of $\mathbb{Z}_n$ on its vertices. This paper concerns 2-cell embeddings of connected circulants on closed orientable surfaces. Embeddings on the sphere (the planar case) were classified by Heuberger (2003), and by a theorem of Thomassen (1991), there are only finitely many vertex-transitive graphs with minimum genus $g$, for any given integer $g \ge 3$. Here we completely determine all connected circulants with minimum genus 1 or 2; this corrects and extends an attempted classification of all toroidal circulants by Costa, Strapasson, Alves and Carlos (2010).</p>2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p29Extensions of Infinite Partition Regular Systems2015-05-22T15:30:28+10:00Neil Hindmannhindman@aol.comImre LeaderLeader@dpmms.cam.ac.ukDona Straussd.strauss@hull.ac.uk<p>A finite or infinite matrix $A$ with rational entries (and only finitely many non-zero entries in each row) is called <em>image partition regular</em> if, whenever the natural numbers are finitely coloured, there is a vector $x$, with entries in the natural numbers, such that $Ax$ is monochromatic. Many of the classicial results of Ramsey theory are naturally stated in terms of image partition regularity.</p><p>Our aim in this paper is to investigate maximality questions for image partition regular matrices. When is it possible to add rows on to $A$ and remain image partition regular? When can one add rows but `nothing new is produced'? What about adding rows and also new variables? We prove some results about extensions of the most interesting infinite systems, and make several conjectures.</p><p>Our most surprising positive result is a compatibility result for Milliken-Taylor systems, stating that (in many cases) one may adjoin one Milliken-Taylor system to a translate of another and remain image partition regular. This is in contrast to earlier results, which had suggested a strong inconsistency between different Milliken-Taylor systems. Our main tools for this are some algebraic properties of $\beta {\mathbb N}$, the Stone-<span>Č</span>ech compactification of the natural numbers.</p>2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p30Fan-Extensions in Fragile Matroids2015-05-22T15:30:49+10:00Carolyn Chuncarolyn.chun@brunel.ac.ukDeborah Chundeborah.chun@mail.wvu.eduDillon Mayhewdillon.mayhew@msor.vuw.ac.nzStefan H. M. Van Zwamsvanzwam@math.lsu.edu<p>If $\mathcal{S}$ is a set of matroids, then the matroid $M$ is $\mathcal{S}$-fragile if, for every element $e\in E(M)$, either $M\backslash e$ or $M/e$ has no minor isomorphic to a member of $\mathcal{S}$. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when $\mathcal{M}$ is a minor-closed class of $\mathcal{S}$-fragile matroids, and $N\in \mathcal{M}$, the only members of $\mathcal{M}$ that contain $N$ as a minor are obtained from $N$ by increasing the length of fans. We prove that if this is the case, then we can certify it with a finite case-analysis. The analysis involves examining matroids that are at most two elements larger than $N$.</p>2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p31When Does the Set of $(a, b, c)$-Core Partitions Have a Unique Maximal Element?2015-05-22T15:31:08+10:00Amol Aggarwalagg_a@mit.eduIn 2007, Olsson and Stanton gave an explicit form for the largest $(a, b)$-core partition, for any relatively prime positive integers $a$ and $b$, and asked whether there exists an $(a, b)$-core that contains all other $(a, b)$-cores as subpartitions; this question was answered in the affirmative first by Vandehey and later by Fayers independently. In this paper we investigate a generalization of this question, which was originally posed by Fayers: for what triples of positive integers $(a, b, c)$ does there exist an $(a, b, c)$-core that contains all other $(a, b, c)$-cores as subpartitions? We completely answer this question when $a$, $b$, and $c$ are pairwise relatively prime; we then use this to generalize the result of Olsson and Stanton.2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p32Containment Game Played on Random Graphs: Another Zig-Zag Theorem2015-05-22T15:31:37+10:00Paweł Prałatpralat@ryerson.caWe consider a variant of the game of Cops and Robbers, called Containment, in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop). The cops win by "containing'' the robber, that is, by occupying all edges incident with a vertex occupied by the robber. The minimum number of cops, $\xi(G)$, required to contain a robber played on a graph $G$ is called the containability number, a natural counterpart of the well-known cop number $c(G)$. This variant of the game was recently introduced by Komarov and Mackey, who proved that for every graph $G$, $c(G) \le \xi(G) \le \gamma(G) \Delta(G)$, where $\gamma(G)$ and $\Delta(G)$ are the domination number and the maximum degree of $G$, respectively. They conjecture that an upper bound can be improved and, in fact, $\xi(G) \le c(G) \Delta(G)$. (Observe that, trivially, $c(G) \le \gamma(G)$.) This seems to be the main question for this game at the moment. By investigating expansion properties, we provide asymptotically almost sure bounds on the containability number of binomial random graphs $\mathcal{G}(n,p)$ for a wide range of $p=p(n)$, showing that it forms an intriguing zigzag shape. This result also proves that the conjecture holds for some range of $p$ (or holds up to a constant or an $O(\log n)$ multiplicative factors for some other ranges).2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p33On a Generalization of Thue Sequences2015-05-22T15:31:55+10:00Jaka Kranjcjaka.kranjc@fis.unm.siBorut Lužarborut.luzar@gmail.comMartina Mockovčiakovámmockov@ntis.zcu.czRoman Sotákroman.sotak@upjs.skA sequence is <em>Thue</em> or <em>nonrepetitive</em> if it does not contain a repetition of any length. We consider a generalization of this notion. A $j$-subsequence of a sequence $S$ is a subsequence in which two consecutive terms are at indices of difference $j$ in $S$. A $k$-<em>Thue sequence</em> is a sequence in which every $j$-subsequence, for $1\le j \le k$, is also Thue. It was conjectured that $k+2$ symbols are enough to construct an arbitrarily long $k$-Thue sequence and shown that the conjecture holds for $k \in \{2,3,5\}$. In this paper we present a construction of $k$-Thue sequences on $2k$ symbols, which improves the previous bound of $2k + 10\sqrt{k}$. Additionaly, we define cyclic $k$-Thue sequences and present a construction of such sequences of arbitrary lengths when $k=2$ using four symbols, with three exceptions. As a corollary, we obtain tight bounds for total Thue colorings of cycles. We conclude the paper with some open problems.2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p34Generating the Cycle Space of Planar Graphs2015-05-22T15:32:10+10:00Matthias Hamannmatthias.hamann@math.uni-hamburg.de<span>We prove that the cycle space of every planar finitely separable 3-connected graph </span><span>$G$ is generated by some </span><span>$\operatorname{Aut}(G)$-invariant nested set of cycles. We also discuss the situation in the case of smaller connectivity.</span>2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p35Symmetric Chain Decompositions of Quotients by Wreath Products2015-05-22T15:32:39+10:00Dwight Duffusdwight@mathcs.emory.eduKyle Thayerkthayer@cs.washington.edu<p>Subgroups of the symmetric group $S_n$ act on $C^n$ (the $n$-fold product $C \times \cdots \times C$ of a chain $C$) by permuting coordinates, and induce automorphisms of the power $C^n$. For certain families of subgroups of $S_n$, the quotients defined by these groups can be shown to have symmetric chain decompositions (SCDs). These SCDs allow us to enlarge the collection of subgroups $G$ of $S_n$ for which the quotient $\mathbf{2}^n/G$ on the Boolean lattice $\mathbf{2}^n$ is a symmetric chain order (SCO). The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.</p>2015-05-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p36The Parametric Frobenius Problem2015-05-22T15:32:59+10:00Bjarke Hammersholt Rounebjarke.roune@gmail.comKevin WoodsKevin.Woods@oberlin.edu<p>Given relatively prime positive integers $a_1,\ldots,a_n$, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the $a_i$. We examine the parametric version of this problem: given $a_i=a_i(t)$ as functions of $t$, compute the Frobenius number as a function of $t$. A function $f:\mathbb{Z}_+\rightarrow\mathbb{Z}$ is a quasi-polynomial if there exists a period $m$ and polynomials $f_0,\ldots,f_{m-1}$ such that $f(t)=f_{t\bmod m}(t)$ for all $t$. We conjecture that, if the $a_i(t)$ are polynomials (or quasi-polynomials) in $t$, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large $t$. We prove this in the case where the $a_i(t)$ are linear functions, and also prove it in the case where $n$ (the number of generators) is at most 3.</p>2015-05-22T00:00:00+10:00