http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2016-12-22T13:33:51+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><p>ISSN: <span lang="EN">1077-8926</span></p>http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p1Isotropic Matroids I: Multimatroids and Neighborhoods2016-10-14T13:52:39+11:00Robert Brijderrobert.brijder@uhasselt.beLorenzo Tralditraldil@lafayette.edu<p>Several properties of the isotropic matroid of a looped simple graph are presented. Results include a characterization of the multimatroids that are associated with isotropic matroids and several ways in which the isotropic matroid of $G$ incorporates information about graphs locally equivalent to $G$. Specific results of the latter type include a characterization of graphs that are locally equivalent to bipartite graphs, a direct proof that two forests are isomorphic if and only if their isotropic matroids are isomorphic, and a way to express local equivalence indirectly, using only edge pivots.</p>2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p2Isotropic Matroids II: Circle Graphs2016-10-14T13:52:39+11:00Robert Brijderrobert.brijder@uhasselt.beLorenzo Tralditraldil@lafayette.edu<p>We present several characterizations of circle graphs, which follow from Bouchet’s circle graph obstructions theorem.</p>2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p3On the Potts Antiferromagnet on Random Graphs2016-10-14T13:52:39+11:00Amin Coja-Oghlancoja-oghlan@mathematik.uni-frankfurt.deNor Jaafarijaafari@math.uni-frankfurt.deExtending a prior result of Contucci et al. [Comm. Math. Phys. 2013], we determine the free energy of the Potts antiferromagnet on the Erdős–Rényi random graph at all temperatures for average degrees $d\le (2k-1)\ln k - 2 - k^{-1/2}$. In particular, we show that for this regime of $d$ there does not occur a phase transition.2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p4Coloring Non-Crossing Strings2016-10-14T13:52:39+11:00Louis EsperetLouis.Esperet@grenoble-inp.frDaniel Gonçalvesdaniel.goncalves@lirmm.frArnaud Labourelarnaud.labourel@lif.univ-mrs.frFor a family of geometric objects in the plane $\mathcal{F}=\{S_1,\ldots,S_n\}$, define $\chi(\mathcal{F})$ as the least integer $\ell$ such that the elements of $\mathcal{F}$ can be colored with $\ell$ colors, in such a way that any two intersecting objects have distinct colors. When $\mathcal{F}$ is a set of pseudo-disks that may only intersect on their boundaries, and such that any point of the plane is contained in at most $k$ pseudo-disks, it can be proven that $\chi(\mathcal{F})\le 3k/2 + o(k)$ since the problem is equivalent to cyclic coloring of plane graphs. In this paper, we study the same problem when pseudo-disks are replaced by a family $\mathcal{F}$ of pseudo-segments (a.k.a. strings) that do not cross. In other words, any two strings of $\mathcal{F}$ are only allowed to "touch" each other. Such a family is said to be $k$-touching if no point of the plane is contained in more than $k$ elements of $\mathcal{F}$. We give bounds on $\chi(\mathcal{F})$ as a function of $k$, and in particular we show that $k$-touching segments can be colored with $k+5$ colors. This partially answers a question of Hliněný (1998) on the chromatic number of contact systems of strings.2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p5Packing Polynomials on Multidimensional Integer Sectors2016-10-14T13:52:39+11:00Luis B. Moraleslbm@unam.mx<p>Denoting the real numbers and the nonnegative integers, respectively, by ${\bf R}$ and ${\bf N}$, let $S$ be a subset of ${\bf N}^n$ for $n = 1, 2,\ldots$, and $f$ be a mapping from ${\bf R}^n$ into ${\bf R}$. We call $f$ a <em>packing function</em> on $S$ if the restriction $f|_{S}$ is a bijection onto ${\bf N}$. For all positive integers $r_1,\ldots,r_{n-1}$, we consider the <em>integer sector </em>\[I(r_1, \ldots, r_{n-1}) =\{(x_1,\ldots,x_n) \in N^n \; | \; x_{i+1} \leq r_ix_i \mbox{ for } i = 1,\ldots,n-1 \}.\] Recently, Melvyn B. Nathanson (2014) proved that for $n=2$ there exist two quadratic packing polynomials on the sector $I(r)$. Here, for $n>2$ we construct $2^{n-1}$ packing polynomials on multidimensional integer sectors. In particular, for each packing polynomial on ${\bf N}^n$ we construct a packing polynomial on the sector $I(1, \ldots, 1)$.</p>2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p6On Matchings in Stochastic Kronecker Graphs2016-10-14T13:52:39+11:00Justyna Banaszaktabor@amu.edu.pl<p>The stochastic Kronecker graph is a random structure whose vertex set is a hypercube and the probability of an edge depends on the structure of its ends. We prove that a.a.s. as soon as the stochastic Kronecker graph becomes connected it contains a perfect matching.</p>2016-10-03T20:47:23+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p7On Almost-Regular Edge Colourings of Hypergraphs2016-10-14T13:52:39+11:00Darryn Bryantdb@maths.uq.edu.au<p>We prove that if ${\cal{H}}=(V({\cal{H}}),{\cal{E}}({\cal{H}}))$ is a hypergraph, $\gamma$ is an edge colouring of ${\cal{H}}$, and $S\subseteq V({\cal{H}})$ such that any permutation of $S$ is an automorphism of ${\cal{H}}$, then there exists a permutation $\pi$ of ${\cal{E}}({\cal{H}})$ such that $|\pi(E)|=|E|$ and $\pi(E)\setminus S=E\setminus S$ for each $E\in{\cal{H}}({\cal{H}})$, and such that the edge colouring $\gamma'$ of ${\cal{H}}$ given by $\gamma'(E)=\gamma(\pi^{-1}(E))$ for each $E\in{\cal{E}}({\cal{H}})$ is almost regular on $S$. The proof is short and elementary. We show that a number of known results, such as Baranyai's Theorem on almost-regular edge colourings of complete $k$-uniform hypergraphs, are easy corollaries of this theorem.</p>2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p8Upper Bounds for Stern's Diatomic Sequence and Related Sequences2016-10-14T13:52:39+11:00Colin Defantcdefant@ufl.edu<pre>Let <span>$(s_2(n))_{n=0}^\infty$</span> denote Stern's diatomic sequence. For <span>$n\geq 2$</span>, we may view <span>$s_2(n)$</span> as the number of partitions of <span>$n-1$</span> into powers of <span>$2$</span> with each part occurring at most twice. More generally, for integers <span>$b,n\geq 2$</span>, let <span>$s_b(n)$</span> denote the number of partitions of <span>$n-1$</span> into powers of <span>$b$</span> with each part occurring at most <span>$b$</span> times. Using this <span>combinatorial</span> interpretation of the sequences <span>$s_b(n)$</span>, we use the transfer-matrix method to develop a means of calculating <span>$s_b(n)$</span> for certain values of <span>$n$</span>. This then allows us to derive upper bounds for <span>$s_b(n)$</span> for certain values of <span>$n$</span>. In the special case <span>$b=2$</span>, our bounds improve upon the current upper bounds for the Stern sequence. In addition, we are able to prove that <span>$\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}$</span>.</pre>2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p9The Smith and Critical Groups of the Square Rook's Graph and its Complement2016-10-14T13:52:39+11:00Joshua E. Duceyduceyje@jmu.eduJonathan Gerhardgerha2jm@dukes.jmu.eduNoah Watsonwatsonnj@dukes.jmu.eduLet $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$. In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement. This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs. In doing so we verify a 1986 conjecture of Rushanan.2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p10All or Nothing at All2016-10-14T13:52:39+11:00Paolo D'Arcopaodar@dia.unisa.itNavid Nasr Esfahaninnasresfahani@uwaterloo.caDouglas R. Stinsondstinson@uwaterloo.caWe continue a study of unconditionally secure all-or-nothing transforms (AONT) begun by Stinson (2001). An AONT is a bijective mapping that constructs $s$ outputs from $s$ inputs. We consider the security of $t$ inputs, when $s-t$ outputs are known. Previous work concerned the case $t=1$; here we consider the problem for general $t$, focussing on the case $t=2$. We investigate constructions of binary matrices for which the desired properties hold with the maximum probability. Upper bounds on these probabilities are obtained via a quadratic programming approach, while lower bounds can be obtained from combinatorial constructions based on symmetric BIBDs and cyclotomy. We also report some results on exhaustive searches and random constructions for small values of $s$.2016-10-05T20:42:42+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p11A Short Conceptual Proof of Narayana's Path-Counting Formula2016-10-14T13:52:39+11:00Mihai Ciucumciucu@indiana.eduWe deduce Narayana's formula for the number of lattice paths that fit in a Young diagram as a direct consequence of the Gessel-Viennot theorem on non-intersecting lattice paths.2016-10-14T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p12Upper Bounds on the Minimum Size of Hamilton Saturated Hypergraphs2016-10-28T12:28:40+11:00Andrzej Rucińskirucinski@amu.edu.plAndrzej Żakzakandrz@agh.edu.plFor $1\leqslant \ell< k$, an <em>$\ell$-overlapping $k$-cycle</em> is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices.<br /><br />A $k$-uniform hypergraph $H$ is <em>$\ell$-Hamiltonian saturated</em> if $H$ does not contain an $\ell$-overlapping Hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let $\mathrm{sat}(n,k,\ell)$ be the smallest number of edges in an $\ell$-Hamiltonian saturated $k$-uniform hypergraph on $n$ vertices. In the case of graphs Clark and Entringer showed in 1983 that $\mathrm{sat}(n,2,1)=\lceil \tfrac{3n}2\rceil$. The present authors proved that for $k\geqslant 3$ and $\ell=1$, as well as for all $0.8k\leqslant \ell\leq k-1$, $\mathrm{sat}(n,k,\ell)=\Theta(n^{\ell})$. In this paper we prove two upper bounds which cover the remaining range of $\ell$. The first, quite technical one, restricted to $\ell\geqslant\frac{k+1}2$, implies in particular that for $\ell=\tfrac23k$ and $\ell=\tfrac34k$ we have $\mathrm{sat}(n,k,\ell)=O(n^{\ell+1})$. Our main result provides an upper bound $\mathrm{sat}(n,k,\ell)=O(n^{\frac{k+\ell}2})$ valid for all $k$ and $\ell$. In the smallest open case we improve it further to $\mathrm{sat}(n,4,2)=O(n^{\frac{14}5})$.2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p13Intersections of the Hermitian Surface with Irreducible Quadrics in Even Characteristic2016-10-28T12:28:52+11:00Angela Agugliaangela.aguglia@poliba.itLuca Giuzziluca.giuzzi@unibs.itWe determine the possible intersection sizes of a Hermitian surface ${\mathcal H}$ with an irreducible quadric of $PG(3,q^2)$ sharing at least a tangent plane at a common non-singular point when $q$ is even.<br /><br />2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p14Triangular Fully Packed Loop Configurations of Excess 22016-10-28T12:29:01+11:00Sabine Beilsabine.beil@univie.ac.atTriangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple $(u,v;w)$ of $01$-words encoding its boundary conditions which must necessarily satisfy that $d(u)+d(v)\leq d(w)$, where $d(u)$ denotes the number of inversions in $u$. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers of FPLs having given link patterns. Later, Wieland drift — a map on TFPLs that is based on Wieland gyration — was defined. The main contribution of this article will be a linear expression for the number of TFPLs with boundary $(u,v;w)$ where $d(w)-d(u)-d(v)=2$ in terms of numbers of stable TFPLs, that is, TFPLs invariant under Wieland drift. This linear expression generalises already existing enumeration results for TFPLs with boundary $(u,v;w)$ where $d(w)-d(u)-d(v)=0,1$.2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p15Some Self-Orthogonal Codes Related to Higman's Geometry2016-10-28T12:29:09+11:00Jamshid Moorijamshid.moori@nwu.ac.zaB. D. Rodriguesrodrigues@ukzn.ac.za<p>We examine some self-orthogonal codes constructed from a rank-5 primitive permutation representation of degree 1100 of the sporadic simple group ${\rm HS}$ of Higman-Sims. We show that ${\rm Aut}(C) = {\rm HS}{:}2$, where $C$ is a code of dimension 21 associated with Higman's geometry.</p>2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p16Incidences with Curves in $\mathbb{R}^d$2016-10-28T12:29:19+11:00Micha Sharirmichas@post.tau.ac.ilAdam Shefferadamsh@gmail.comNoam Solomonnoam.solom@gmail.com<div>We prove that the number of incidences between $m$ points and $n$ bounded-degree curves with $k$ degrees of freedom in ${\mathbb R}^d$ is</div><div>\[ O\left(m^{\frac{k}{dk-d+1}+\varepsilon}n^{\frac{dk-d}{dk-d+1}}+ \sum_{j=2}^{d-1} m^{\frac{k}{jk-j+1}+\varepsilon}n^{\frac{d(j-1)(k-1)}{(d-1)(jk-j+1)}} q_j^{\frac{(d-j)(k-1)}{(d-1)(jk-j+1)}}+m+n\right),\]</div><div>for any $\varepsilon>0$, where the constant of proportionality depends on $k, \varepsilon$ and $d$, provided that no $j$-dimensional surface of degree $\le c_j(k,d,\varepsilon)$, a constant parameter depending on $k$, $d$, $j$, and $\varepsilon$, contains more than $q_j$ input curves, and that the $q_j$'s satisfy certain mild conditions. </div><div><p>This bound generalizes the well-known planar incidence bound of Pach and Sharir to $\mathbb{R}^d$. It generalizes a recent result of Sharir and Solomon concerning point-line incidences in four dimensions (where d=4 and k=2), and partly generalizes a recent result of Guth (as well as the earlier bound of Guth and Katz) in three dimensions (Guth's three-dimensional bound has a better dependency on $q_2$). It also improves a recent d-dimensional general incidence bound by Fox, Pach, Sheffer, Suk, and Zahl, in the special case of incidences with algebraic curves. Our results are also related to recent works by Dvir and Gopi and by Hablicsek and Scherr concerning rich lines in high-dimensional spaces. Our bound is not known to be tight in most cases.</p></div>2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p17Grasshopper Avoidance of Patterns2016-10-28T12:29:26+11:00Michał Dębskimichaldebski87@gmail.comUrszula Pastwaurszula@pastwa.plKrzysztof Węsekk.wesek@mini.pw.edu.plMotivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern $p\in E^+$ <em>occurs with jumps</em> in a word $w=a_1a_2\ldots a_k \in A^+$, if there exist a non-erasing morphism $f$ from $E^*$ to $A^*$ and a sequence $(i_1, i_2, \ldots , i_l)$ satisfying $i_{j+1}\in\{ i_j+1, i_j+2 \}$ for $j=1, 2, \ldots, l-1$, such that $f(p) = a_{i_1}a_{i_2}\ldots a_{i_l}.$ For example, a pattern $xx$ occurs with jumps in a word $abdcadbc$ (for $x \mapsto abc$). A pattern $p$ is <em>grasshopper $k$-avoidable</em> if there exists an alphabet $A$ of $k$ elements, such that there exist arbitrarily long words over $A$ in which $p$ does not occur with jumps. The minimal such $k$ is the <em>grasshopper avoidability index</em> of $p$. It appears that this notion is related to two other problems: pattern avoidance on graphs and pattern-free colorings of the Euclidean plane. In particular, we show that a sequence avoiding a pattern $p$ with jumps can be a tool to construct a line $p$-free coloring of $\mathbb{R}^2$.<br /> <br />In our work, we determine the grasshopper avoidability index of patterns $\alpha^n$ for all $n$ except $n=5$. We also show that every doubled pattern is grasshopper $(2^7+1)$-avoidable, every pattern on $k$ variables of length at least $2^k$ is grasshopper $37$-avoidable, and there exists a constant $c$ such that every pattern of length at least $c$ on $2$ variables is grasshopper $3$-avoidable (those results are proved using the entropy compression method).2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p18Modification of Griffiths' Result for Even Integers2016-10-28T12:29:34+11:00Eshita Mazumdartuli.mazumdar@gmail.comSneh Bala Sinhasneh@isid.ac.inFor a finite abelian group $G$ with $\exp(G)=n$, the arithmetical invariant $\mathsf s_A(G)$ is defined to be the least integer $k$ such that any sequence $S$ with length $k$ of elements in $G$ has a $A$ weighted zero-sum subsequence of length $n$. When $A=\{1\}$, it is <em>the Erdős-Ginzburg-Ziv constant</em> and is denoted by $\mathsf s (G)$. For certain class of sets $A$, we already have some general bounds for these weighted constants corresponding to the cyclic group $\mathbb{Z}_n$, which was given by Griffiths. For odd integer $n$, Adhikari and Mazumdar generalized the above mentioned results in the sense that they hold for more sets $A$. In the present paper we modify Griffiths' method for even $n$ and obtain general bound for the weighted constants for certain class of weighted sets which include sets that were not covered by Griffiths for $n\equiv 0 \pmod{4}$.2016-10-28T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p19Every Graph $G$ is Hall $\Delta(G)$-Extendible2016-11-10T13:06:07+11:00Sarah Hollidayshollid4@kennesaw.eduJennifer Vandenbusschejvandenb@kennesaw.eduErik E. Westlundewestlun@kennesaw.edu<p>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 $G$ with list assignment $L$, abbreviated $(G,L)$, satisfies <em>Hall's condition</em> if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ 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 $k$-extendible</em> for some $k \geq \chi(G)$ if every $k$-precoloring of $G$ whose corresponding list assignment is Hall can be extended to a proper $k$-coloring of $G$. In 2011, Bobga et al. posed the question: If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-extendible? This paper establishes an affirmative answer to this question: every graph $G$ is Hall $\Delta(G)$-extendible. Results relating to the behavior of Hall extendibility under subgraph containment are also given. Finally, for certain graph families, the complete spectrum of values of $k$ for which they are Hall $k$-extendible is presented. We include a focus on graphs which are Hall $k$-extendible for all $k \geq \chi(G)$, since these are graphs for which satisfying the obviously necessary Hall's condition is also sufficient for a precoloring to be extendible.</p>2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p20The Cycle Descent Statistic on Permutations2016-11-10T13:08:09+11:00Jun Mamajun904@sjtu.edu.cnShi-Mei Mashimeimapapers@163.comYeong-Nan Yehmayeh@math.sinica.edu.twXu Zhus_j_z_x@sjtu.edu.cnIn this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect matchings.2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p21Hadwiger's Conjecture for 3-Arc Graphs2016-11-10T13:09:37+11:00David Wooddavid.wood@monash.eduGuangjun Xugjxu11@gmail.comSanming Zhousmzhou@ms.unimelb.edu.au<p>The 3-arc graph of a digraph $D$ is defined to have vertices the arcs of $D$ such that two arcs $uv, xy$ are adjacent if and only if $uv$ and $xy$ are distinct arcs of $D$ with $v\ne x$, $y\ne u$ and $u,x$ adjacent. We prove Hadwiger's conjecture for 3-arc graphs.</p>2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p22Preserving the Number of Cycles of Length $k$ in a Growing Uniform Permutation2016-11-10T13:11:12+11:00Philippe Duchonduchon@labri.frRomaric Duvignauromaric.duvignau@lif.univ-mrs.frThe goal of this work is to describe a uniform generation tree for permutations which preserves the number of $k$-cycles between any permutation (except for a small unavoidable subset of optimal size) of the tree and its direct children. Moreover, the tree we describe has the property that if the number of $k$-cycles does not change during any $k$ consecutive levels, then any further random descent will always yield permutations with that same number of $k$-cycles. This specific additional property yields interesting applications for exact sampling. We describe a new random generation algorithm for permutations with a fixed number of $k$-cycles in $n+\mathcal{O}(1)$ expected calls to a random integer sampler. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter $1/k$.2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p23Symmetric Isostatic Frameworks with $\ell^1$ or $\ell^\infty$ Distance Constraints2016-11-10T13:12:52+11:00Derek Kitsond.kitson@lancaster.ac.ukBernd Schulzeb.schulze@lancaster.ac.ukCombinatorial characterisations of minimal rigidity are obtained for symmetric $2$-dimensional bar-joint frameworks with either $\ell^1$ or $\ell^\infty$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p24Resolvable Group Divisible Designs with Large Groups2016-11-10T13:14:03+11:00Peter J. Dukesdukes@uvic.caEsther R. Lamkenlamken@caltech.eduAlan C.H. Lingaling@emba.uvm.edu<p>We prove that the necessary divisibility conditions are sufficient for the existence of resolvable group divisible designs with a fixed number of sufficiently large groups. Our method combines an application of the Rees product construction with a streamlined recursion based on incomplete transversal designs. With similar techniques, we also obtain new results on decompositions of complete multipartite graphs into a prescribed graph.</p>2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p25Invariant Tensors and the Cyclic Sieving Phenomenon2016-11-10T13:16:01+11:00Bruce W. WestburyBruce.Westbury@warwick.ac.ukWe construct a large class of examples of the cyclic sieving phenomenon by exploiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p26Corners in Tree-Like Tableaux2016-11-10T13:17:19+11:00Paweł Hitczenkophitczenko@math.drexel.eduAmanda Lohssagp47@drexel.edu<p>In this paper, we study tree<span>–</span>like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree<span>–</span>like tableaux and the total number of corners in symmetric tree<span>–</span>like tableaux. In this paper, we prove both conjectures. Our proofs are based off of the bijection with permutation tableaux or type<span>–</span>B permutation tableaux and consequently, we also prove results for these tableaux. In addition, we derive the limiting distribution of the number of occupied corners in random tree<span>–</span>like tableaux and random symmetric tree<span>–</span>like tableaux.</p>2016-11-10T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p27A Note on $\mathtt{V}$-free 2-matchings2016-11-25T12:18:45+11:00Kristóf Bércziberkri@cs.elte.huAttila Bernáthbernath@cs.elte.huMáté Vizervizermate@gmail.comMotivated by a conjecture of Liang, we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of the hypergraph matching problem. We close the paper by showing that it is NP-complete to decide whether one of the color classes of a bipartite graph can be covered by a $\mathtt{V}$-free $2$-matching.2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p28On Universal Hypergraphs2016-11-25T12:20:00+11:00Samuel Hetterichhetterich@math.uni-frankfurt.deOlaf ParczykParczyk@math.uni-frankfurt.deYury Personperson@math.uni-frankfurt.de<pre><!--StartFragment-->A <span>hypergraph</span> <span>$H$</span> is called universal for a family <span>$\mathcal{F}$</span> of <span>hypergraphs</span>, if it contains every <span>hypergraph</span> <span>$F \in \mathcal{F}$</span> as a copy. For the family of $r$-uniform hypergraphs with maximum vertex degree bounded by $\Delta$ and at most $n$ vertices any universal hypergraph has to contain $\Omega(n^{r-r/\Delta})$ many edges. We exploit constructions of Alon and Capalbo to obtain universal $r$-uniform hypergraphs with the optimal number of edges $O(n^{r-r/\Delta})$ when $r$ is even, $r \mid \Delta$ or $\Delta=2$. Further we generalize the result of Alon and Asodi about optimal universal graphs for the family of graphs with at most $m$ edges and no isolated vertices to hypergraphs.</pre>2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p29Inclusion Matrices and the MDS Conjecture2016-11-25T12:22:39+11:00Ameera Chowdhuryameerah@alumni.caltech.eduLet $\mathbb{F}_{q}$ be a finite field of order $q$ with characteristic $p$. An arc is an ordered family of at least $k$ vectors in $\mathbb{F}_{q}^{k}$ in which every subfamily of size $k$ is a basis of $\mathbb{F}_{q}^{k}$. The MDS conjecture, which was posed by Segre in 1955, states that if $k \leq q$, then an arc in $\mathbb{F}_{q}^{k}$ has size at most $q+1$, unless $q$ is even and $k=3$ or $k=q-1$, in which case it has size at most $q+2$. <br /><br />We propose a conjecture which would imply that the MDS conjecture is true for almost all values of $k$ when $q$ is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when $k \leq p$, and if $q$ is not prime, for $k \leq 2p-2$. To accomplish this, given an arc $G \subset \mathbb{F}_{q}^{k}$ and a nonnegative integer $n$, we construct a matrix $M_{G}^{\uparrow n}$, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix $M_{G}^{\uparrow n}$ to properties of the arc $G$ and may provide new tools in the computational classification of large arcs.<br /><br />2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p30Antipode Formulas for some Combinatorial Hopf Algebras2016-11-25T12:23:16+11:00Rebecca Patriaspatriasr@lacim.caMotivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p31Equitable Orientations of Sparse Uniform Hypergraphs2016-11-25T12:24:01+11:00Nathann Cohennathann.cohen@gmail.comWilliam Lochetwilliam.lochet@gmail.com<p>Caro, West, and Yuster (2011) studied how $r$-uniform hypergraphs can be oriented in such a way that (generalizations of) indegree and outdegree are as close to each other as can be hoped. They conjectured an existence result of such orientations for sparse hypergraphs, of which we present a proof.</p>2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p32$(s,t)$-Cores: a Weighted Version of Armstrong’s Conjecture2016-11-25T12:25:03+11:00Matthew Fayersm.fayers@qmul.ac.ukThe study of core partitions has been very active in recent years, with the study of <em>$(s,t)$-cores</em> — partitions which are both $s$- and $t$-cores %mdash; playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang.<br /><br />In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525—1539], studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core.<br /><br />2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p33On Total Positivity of Catalan-Stieltjes Matrices2016-11-25T12:25:33+11:00Qiongqiong Panqiong-qiong.pan@etu.univ-lyon1.frJiang Zengzeng@math.univ-lyon1.fr<p>Recently Chen-Liang-Wang (Linear Algerbra Appl. <strong>471</strong> (2015) 383—393) proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix $A$ we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of $A$ is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Catalan-Stieltjes matrix $A$.</p>2016-11-25T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p34Dominating Sequences in Grid-Like and Toroidal Graphs2016-12-09T16:50:27+11:00Boštjan BrešarBostjan.Bresar@um.siCsilla Bujtásbujtas@dcs.uni-pannon.huTanja Gologranctanja.gologranc@gmail.comSandi Klavžarsandi.klavzar@fmf.uni-lj.siGašper Košmrljgkosmrlj@gmail.comBalázs Patkóspatkosb@gmail.comZsolt Tuzatuza@dcs.uni-pannon.huMáté Vizervizermate@gmail.comA longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p35Directed Rooted Forests in Higher Dimension2016-12-09T16:52:03+11:00Olivier Bernardibernardi@brandeis.eduCaroline J. Klivansklivans@brown.eduFor a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p36Large Deviations for Permutations Avoiding Monotone Patterns2016-12-09T16:54:06+11:00Neal Madrasmadras@mathstat.yorku.caLerna Pehlivanlpehlivan@mta.caFor a given permutation $\tau$, let $P_N^{\tau}$ be the uniform probability distribution on the set of $N$-element permutations $\sigma$ that avoid the pattern $\tau$. For $\tau=\mu_k:=123\ldots k$, we consider $P_N^{\mu_k}(\sigma_I=J)$ where $I\sim \gamma N$ and $J\sim \delta N$ for $\gamma, \delta \in (0,1)$. If $\gamma+ \delta \neq 1$, then we are in the large deviations regime with the probability decaying exponentially, and we calculate the limiting value of $P_N^{\mu_k}(\sigma_I=J)^{1/N}$. We also observe that for $\tau = \lambda_{k,\ell} := 12\ldots\ell k(k-1)\ldots(\ell+1)$ and $\gamma+\delta<1$, the limit of $P_N^{\tau}(\sigma_I=J)^{1/N}$ is the same as for $\tau=\mu_k$.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p37The CI Problem for Infinite Groups2016-12-09T16:55:44+11:00Joy Morrisjoy.morris@uleth.caA finite group $G$ is a DCI-group if, whenever $S$ and $S'$ are subsets of $G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S')$ isomorphic, there exists an automorphism $\varphi$ of $G$ with $\varphi(S)=S'$. It is a CI-group if this condition holds under the restricted assumption that $S=S^{-1}$. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CI$_f$-group if the same condition holds under the restricted assumption that $S$ is finite; and an infinite group is a (D)CI$_f$-group if the same condition holds whenever $S$ is both finite and generates $G$.<br /><br />We prove that an infinite (D)CI-group must be a torsion group that is not locally-finite. We find infinite families of groups that are (D)CI$_f$-groups but not strongly (D)CI$_f$-groups, and that are strongly (D)CI$_f$-groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on $\mathbb Z^n$. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI-group exists. <br /><br />2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p38Client-Waiter Games on Complete and Random Graphs2016-12-09T16:56:46+11:00Oren Deanorendean@mail.tau.ac.ilMichael Krivelevichkrivelev@post.tau.ac.ilFor a graph $ G $, a monotone increasing graph property $ \mathcal{P} $ and positive integer $ q $, we define the Client-Waiter game to be a two-player game which runs as follows. In each turn Waiter is offering Client a subset of at least one and at most $ q+1 $ unclaimed edges of $ G $ from which Client claims one, and the rest are claimed by Waiter. The game ends when all the edges have been claimed. If Client's graph has property $ \mathcal{P} $ by the end of the game, then he wins the game, otherwise Waiter is the winner. In this paper we study several Client-Waiter games on the edge set of the complete graph, and the $ H $-game on the edge set of the random graph. For the complete graph we consider games where Client tries to build a large star, a long path and a large connected component. We obtain lower and upper bounds on the critical bias for these games and compare them with the corresponding Waiter-Client games and with the probabilistic intuition. For the $ H $-game on the random graph we show that the known results for the corresponding Maker-Breaker game are essentially the same for the Client-Waiter game, and we extend those results for the biased games and for trees.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p39On the Number of Similar Instances of a Pattern in a Finite Set2016-12-09T16:59:17+11:00Bernardo M. Ábregobernardo.abrego@csun.eduSilvia Fernández-Merchantsilvia.fernandez@csun.eduDaniel J. Katzdaniel.katz@csun.eduLevon Kolesnikovlevon.kolesnikov.518@my.csun.eduNew bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p40Pathwidth and Nonrepetitive List Coloring2016-12-09T17:03:09+11:00Adam Gągolgagol@tcs.uj.edu.plGwenaël Joretgjoret@ulb.ac.beJakub Kozikjkozik@tcs.uj.edu.plPiotr Micekmicek@tcs.uj.edu.plA vertex coloring of a graph is nonrepetitive if there is no path in the graph whose first half receives the same sequence of colors as the second half. While every tree can be nonrepetitively colored with a bounded number of colors (4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently showed that this does not extend to the list version of the problem, that is, for every $\ell \geq 1$ there is a tree that is not nonrepetitively $\ell$-choosable. In this paper we prove the following positive result, which complements the result of Fiorenzi et al.: There exists a function $f$ such that every tree of pathwidth $k$ is nonrepetitively $f(k)$-choosable. We also show that such a property is specific to trees by constructing a family of pathwidth-2 graphs that are not nonrepetitively $\ell$-choosable for any fixed $\ell$.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p41Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras2016-12-09T17:06:53+11:00John W Lawsonj.w.lawson@durham.ac.ukCluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p42Exponential Domination in Subcubic Graphs2016-12-09T17:08:10+11:00Stéphane Bessystephane.bessy@lirmm.frPascal Ochempascal.ochem@lirmm.frDieter Rautenbachdieter.rautenbach@uni-ulm.de<p>As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertices. More precisely, if $S$ is a set of vertices of a graph $G$, then $S$ is an exponential dominating set of $G$ if $\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where ${\rm dist}_{(G,S)}(u,v)$ is the distance between $u\in V(G)\setminus S$ and $v\in S$ in the graph $G-(S\setminus \{ v\})$. The exponential domination number $\gamma_e(G)$ of $G$ is the minimum order of an exponential dominating set of $G$.</p><p style="-qt-paragraph-type: empty; -qt-block-indent: 0; text-indent: 0px; margin: 0px;">In the present paper we study exponential domination in subcubic graphs. Our results are as follows: If $G$ is a connected subcubic graph of order $n(G)$, then $$\frac{n(G)}{6\log_2(n(G)+2)+4}\leq \gamma_e(G)\leq \frac{1}{3}(n(G)+2).$$ For every $\epsilon>0$, there is some $g$ such that $\gamma_e(G)\leq \epsilon n(G)$ for every cubic graph $G$ of girth at least $g$. For every $0<\alpha<\frac{2}{3\ln(2)}$, there are infinitely many cubic graphs $G$ with $\gamma_e(G)\leq \frac{3n(G)}{\ln(n(G))^{\alpha}}$. If $T$ is a subcubic tree, then $\gamma_e(T)\geq \frac{1}{6}(n(T)+2).$ For a given subcubic tree, $\gamma_e(T)$ can be determined in polynomial time. The minimum exponential dominating set problem is APX-hard for subcubic graphs.</p>2016-12-09T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p43Pieri Rules for Classical Groups and Equinumeration between Generalized Oscillating Tableaux and Semistandard Tableaux2016-12-22T13:32:59+11:00Soichi Okadaokada@math.nagoya-u.ac.jp<p>We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretic proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.</p>2016-12-23T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p44The Lattice of Definable Equivalence Relations in Homogeneous $n$-Dimensional Permutation Structures2016-12-22T13:33:26+11:00Samuel Braunfeldswb52@math.rutgers.edu<p>In <em>Homogeneous permutations</em>, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous $n$-dimensional permutation structures (homogeneous structures with $n$ linear orders) for all finite $n$. We prove here that the lattice of $\emptyset$-definable equivalence relations in such a structure can be any finite distributive lattice, providing many new imprimitive examples of homogeneous finite dimensional permutation structures. We conjecture that the distributivity of the lattice of $\emptyset$-definable equivalence relations is necessary, and prove this under the assumption that the reduct of the structure to the language of $\emptyset$-definable equivalence relations is homogeneous. Finally, we conjecture a classification of the primitive examples, and confirm this in the special case where all minimal forbidden structures have order 2.</p><p> </p>2016-12-23T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p45A Sharp Bound for the Product of Weights of Cross-Intersecting Families2016-12-22T13:33:38+11:00Peter Borgpeter.borg@um.edu.mt<p>Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be <em>cross-intersecting</em> if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of subsets of $\{1, \dots, n\}$ of size at most $k$, and let $\mathcal{S}_{n,k}$ denote the family of sets in ${[n] \choose \leq k}$ that contain $1$. The author recently showed that if $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then $|\mathcal{A}||\mathcal{B}| \leq \mathcal{S}_{m,r}||\mathcal{S}_{n,s}|$. We prove a version of this result for the more general setting of \emph{weighted} sets. We show that if $g : {[m] \choose \leq r} \rightarrow \mathbb{R}^+$ and $h : {[n] \choose \leq s} \rightarrow \mathbb{R}^+$ are functions that obey certain conditions, $\mathcal{A} \subseteq {[m] \choose \leq r}$, $\mathcal{B} \subseteq {[n] \choose \leq s}$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then <br />$$\sum_{A \in \mathcal{A}} g(A) \sum_{B \in \mathcal{B}} h(B) \leq \sum_{C \in \mathcal{S}_{m,r}} g(C) \sum_{D \in \mathcal{S}_{n,s}} h(D).$$<br />The bound is attained by taking $\mathcal{A} = \mathcal{S}_{m,r}$ and $\mathcal{B} = \mathcal{S}_{n,s}$. We also show that this result yields new sharp bounds for the product of sizes of cross-intersecting families of integer sequences and of cross-intersecting families of multisets.</p>2016-12-23T00:00:00+11:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i4p46Induced Colorful Trees and Paths in Large Chromatic Graphs2016-12-22T13:33:51+11:00Andras Gyarfasgyarfasa@gmail.comGabor Sarkozygsarkozy@cs.wpi.edu<p>In a proper vertex coloring of a graph a subgraph is <em>colorful</em> if its vertices are colored with different colors. It is well-known (see for example in Gy<span>á</span>rf<span>á</span>s (1980)) that in every proper coloring of a $k$-chromatic graph there is a colorful path $P_k$ on $k$ vertices. The first author proved in 1987 that $k$-chromatic and <em>triangle-free</em> graphs have a path $P_k$ which is an <em>induced subgraph</em>. N.R. Aravind conjectured that these results can be put together: in every proper coloring of a $k$-chromatic triangle-free graph, there is an induced colorful $P_k$. Here we prove the following weaker result providing some evidence towards this conjecture: For a suitable function $f(k)$, in any proper coloring of an $f(k)$-chromatic graph of girth at least five, there is an induced colorful path on $k$ vertices.</p>2016-12-23T00:00:00+11:00