http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2017-11-03T12:11:51+11:00Matt Beckmattbeck@sfsu.eduOpen Journal Systems<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/v24i4p1The Gonality Sequence of Complete Graphs2017-10-05T15:02:18+11:00Filip Coolsf.cools@kuleuven.beMarta Panizzutpanizzut@math.tu-berlin.deThe gonality sequence $(\gamma_r)_{r\geq1}$ of a finite graph/metric graph/algebraic curve comprises the minimal degrees $\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\gamma_r = kd - h$ if $r<g=\frac{(d-1)(d-2)}{2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \frac{k(k+3)}{2} - h$ with $1\leq k\leq d-3$ and $0 \leq h \leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs.2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p2Minimum Cuts of Distance-Regular Digraphs2017-10-05T15:02:18+11:00Saleh Ashkbooss.ashkboos@ec.iut.ac.irGholamreza Omidiromidi@cc.iut.ac.irFateme Shafieifateme.shafiei@math.iut.ac.irKhosro Tajbakhshkhtajbakhsh@modares.ac.ir<p>In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph $\Gamma$, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of $\Gamma$ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular-digraph $\Gamma$ with diameter $D=2$, degree $k\leq 3$ or $\lambda=0$ ($\lambda$ is the number of 2-paths from $u$ to $v$ for an edge $uv$ of $\Gamma$) is super vertex-connected, that is, any minimum vertex cut of $\Gamma$ is the set of all out-neighbors (or in-neighbors) of a single vertex in $\Gamma$. These results extend the same known results for the undirected case with quite different proofs.</p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p3A Major-Index Preserving Map on Fillings2017-10-05T15:02:18+11:00Per Alexanderssonper.w.alexandersson+ejc@gmail.comMehtaab Sawhneymsawhney98@yahoo.comWe generalize a map by S. Mason regarding two combinatorial models for key polynomials, in a way that accounts for the major index. Furthermore we define a similar variant of this map, that regards alternative models for the modified Macdonald polynomials at t=0, and thus partially answers a question by J. Haglund. These maps together imply a certain uniqueness property regarding inversion–and coinversion-free fillings. These uniqueness properties allow us to generalize the notion of charge to a non-symmetric setting, thus answering a question by A. Lascoux and the analogous question in the symmetric setting proves a conjecture by K. Nelson.2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p4Schur-Concavity for Avoidance of Increasing Subsequences in Block-Ascending Permutations2017-10-05T15:02:18+11:00Evan Chenevanchen@mit.eduFor integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1,\dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and which avoids the pattern $12\dots(k+2)$. We exhibit some bijections between such sets, most notably showing that $\# \mathcal L_{k+2} (a_1, \dots, a_n)$ is symmetric in the $a_i$ and is in fact Schur-concave. This generalizes a set of equivalences observed by Mei and Wang.2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p5Disproof of a Conjecture of Neumann-Lara2017-10-05T15:02:18+11:00Bernardo Llanollano@xanum.uam.mxMika Olsenolsen@correo.cua.uam.mx<p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">We disprove the following conjecture due to Víctor Neumann-Lara: for every pair $(r,s)$ of integers such that $r\geq s\geq 2$, there is an infinite set of circulant tournaments $T$ such that the dichromatic number and the cyclic triangle free disconnection of $T$ are equal to $r$ and $s$, respectively. Let $\mathcal{F}_{r,s}$ denote the set of circulant tournaments $T$ with $dc(T)=r$ and $\overrightarrow{\omega }_{3}\left( T\right) =s$. We show that for every integer $s\geq 4$ there exists a lower bound $b(s)$ for the dichromatic number $r$ such that $\mathcal{F}_{r,s}=\emptyset $ for every $r<b(s)$. We construct an infinite set of circulant tournaments $T$ such that $dc(T)=b(s)$ and $\overrightarrow{\omega }_{3}(T)=s$ and give an upper bound $B(s)$ for the dichromatic number $r$ such that for every $r\geq B(s)$ there exists an infinite set $\mathcal{F}_{r,s}$ of circulant tournaments. Some infinite sets $\mathcal{F}_{r,s}$ of circulant tournaments are given for $b(s)<r<B(s)$.</p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p6A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-Polymer2017-10-05T15:02:18+11:00Yuchen Peibaconp@gmail.comIn Matveev-Petrov (2017) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$RSK) was introduced.In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a $q$-polymer model based on the $q$RSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and $q$-geometric weights.We use the $q$-local moves to define a generalisation of the $q$RSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the $q$-polymer in $q$-geometric environment, formulate a $q$-version of the multilayer polynuclear growth model ($q$PNG) and write down the joint distribution of the $q$-polymer partition functions at a fixed time.2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p7A Short Proof of Moll's Minimal Conjecture2017-10-05T15:02:18+11:00Lun Lvklunlv@163.comWe give a short proof of Moll's minimal conjecture, which has been confirmed by Chen and Xia.2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p8Local Resilience for Squares of Almost Spanning Cycles in Sparse Random Graphs2017-10-05T15:02:18+11:00Andreas Noeveranoever@inf.ethz.chAngelika Stegersteger@inf.ethz.ch<p><!--StartFragment-->In 1962, Pósa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Komlós, Sárkőzy, and Szemerédi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\epsilon > 0$ and $p=n^{-1/2+\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a <em>spanning</em> cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c<2/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1/3+c)n$ vertices.</p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p9Smaller Subgraphs of Minimum Degree $k$2017-10-05T15:02:18+11:00Frank Moussetmoussetf@inf.ethz.chAndreas Noeveranoever@inf.ethz.chNemanja Škorićnskoric@inf.ethz.ch<p>In 1990 Erdős, Faudree, Rousseau and Schelp proved that for $k \ge 2$, every graph with $n \ge k+1$ vertices and $(k-1)(n-k+2)+\binom{k-2}{2}+1$ edges contains a subgraph of minimum degree $k$ on at most $n-\sqrt{n/6k^3}$ vertices. They conjectured that it is possible to remove at least $\epsilon_k n$ many vertices and remain with a subgraph of minimum degree $k$, for some $\epsilon_k>0$. We make progress towards their conjecture by showing that one can remove at least order of $\Omega(n/\log n)$ many vertices.</p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p10Fair Representation in the Intersection of Two Matroids2017-10-05T15:02:18+11:00Ron Aharoniraharoni@gmail.comEli Bergerberger.haifa@gmail.comDani Kotlardannykotlar@gmail.comRan Zivranzivziv@gmail.com<p>Mysteriously, hypergraphs that are the intersection of two matroids behave in some respects almost as well as one matroid. In the present paper we study one such phenomenon - the surprising ability of the intersection of two matroids to fairly represent the parts of a given partition of the ground set. For a simplicial complex $\mathcal{C}$ denote by $\beta(\mathcal{C})$ the minimal number of edges from $\mathcal{C}$ needed to cover the ground set. If $\mathcal{C}$ is a matroid then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in \mathcal{C}$ meeting each $A_i$ in at least $\lfloor \frac{|A_i|}{\beta(\mathcal{C})}\rfloor$ elements. We conjecture that a slightly weaker result is true for the intersection of two matroids: if $\mathcal{D}=\mathcal{P}\cap \mathcal{Q}$, where $\mathcal{P},\mathcal{Q}$ are matroids on the same ground set $V$ and $\beta(\mathcal{P}), \beta(\mathcal{Q}) \le k$, then for every partition $A_1, \ldots, A_m$ of the ground set there exists a set $S \in \mathcal{D}$ meeting each $A_i$ in at least $\frac{1}{k}|A_i|-1$ elements. We prove that if $m=2$ (meaning that the partition is into two sets) there is a set belonging to $\mathcal{D}$ meeting each $A_i$ in at least $(\frac{1}{k}-\frac{1}{|V|})|A_i|-1$ elements.</p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p11The Universal Gröbner Basis of a Binomial Edge Ideal2017-10-05T15:02:18+11:00Mourtadha Badianem.badiane1@nuigalway.ieIsaac Burkei.burke1@outlook.comEmil Sköldbergemil.skoldberg@nuigalway.ie<p><!--StartFragment-->We show that the universal Gröbner basis and the Graver basis of a binomial edge ideal coincide. We provide a description for this basis set in terms of certain paths in the underlying graph. We conjecture a similar result for a parity binomial edge ideal and prove this conjecture for the case when the underlying graph is the complete graph.<!--EndFragment--></p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p12On Stacked Triangulated Manifolds2017-10-05T15:02:19+11:00Basudeb Dattadattab@math.iisc.ernet.inSatoshi Murais-murai@ist.osaka-u.ac.jpWe prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d \geq 4$, if $\Delta$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1 < i < d-1$, then $\Delta$ is a stacked triangulation of a manifold. These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.<p style="-qt-paragraph-type: empty; -qt-block-indent: 0; text-indent: 0px; margin: 0px;"> </p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p13Longest Monotone Subsequences and Rare Regions of Pattern-Avoiding Permutations2017-10-05T15:02:19+11:00Neal Madrasmadras@mathstat.yorku.caGökhan Yıldırımgkhnyildirim@gmail.comWe consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permutations.<p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;"> </p><p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;"> </p><p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">In addition, for a pattern $\tau$ of length $k$, we scale the plot of a random $\tau$-avoiding permutation down to the unit square and study the "rare region", which is the part of the square that is exponentially unlikely to contain any points. We prove that when $\tau_1>\tau_k$, the complement of the rare region is a closed set that contains the main diagonal of the unit square. For the case $\tau_1=k,$ we also show that the lower boundary of the part of the rare region above the main diagonal is a curve that is Lipschitz continuous and strictly increasing on $[0,1]$.</p>2017-10-06T00:00:00+11:00Copyright (c) http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p14A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group2017-10-19T21:12:42+11:00Alexander Lazaralexander.leo.lazar@gmail.comCartwright (2015) introduced the notion of a weak tropical complex in order to generalize the theory of divisors on graphs from Baker and Norine (2007). A weak tropical complex $\Gamma$ is a $\Delta$-complex equipped with algebraic data that allows it to be viewed as the dual complex to a certain kind of degeneration over a discrete valuation ring. Every graph has a unique tropical complex structure (which is the same structure studied by Baker and Norine) in which divisors correspond to states in the chip-firing game on that graph. Let $G$ and $H$ be graphs, and let $\Gamma$ be a triangulation of $G\times H$ obtained by adding in one diagonal of each resulting square. There is a particular weak tropical complex structure on $\Gamma$ that Cartwright conjectured was closely related to the weak tropical complex structures on $G$ and $H$. The main result of this paper is a proof of Cartwright's conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p15Even Cycles and Even 2-Factors in the Line Graph of a Simple Graph2017-10-19T21:12:49+11:00Arrigo Bonisoliarrigo.bonisoli@unimore.itSimona Bonvicinisimona.bonvicini@unimore.it<p>Let $G$ be a connected graph with an even number of edges. We show that if the subgraph of $G$ induced by the vertices of odd degree has a perfect matching, then the line graph of $G$ has a $2$-factor whose connected components are cycles of even length (an even $2$-factor). For a cubic graph $G$, we also give a necessary and sufficient condition so that the corresponding line graph $L(G)$ has an even cycle decomposition of index $3$, i.e., the edge-set of $L(G)$ can be partitioned into three $2$-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index $m$ in $2d$-regular graphs is also addressed.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p16Quantum Walks on Generalized Quadrangles2017-10-19T21:14:40+11:00Chris Godsilcgodsil@uwaterloo.caKrystal Guoguo.krystal@gmail.comTor G. J. Myklebusttmyklebu@uwaterloo.caWe study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We probabilistically compute the spectrum of the line intersection graphs of two non-isomorphic generalized quadrangles of order $(5^2,5)$ under this matrix and thus provide strongly regular counter-examples to the conjecture.2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p17Covers of D-Type Artin Groups2017-10-19T21:15:08+11:00Meirav Amrammeirav@macs.biu.ac.ilRobert Shwartzrobertsh@ariel.ac.ilMina Teicherteicher@macs.biu.ac.il<p>We study certain quotients of generalized Artin groups which have a natural map onto D-type Artin groups, where the generalized Artin group $A(T)$ is defined by a signed graph $T$. Then we find a certain quotient $G(T)$ according to the graph $T$, which also have a natural map onto $A(D_n)$. We prove that $G(T)$ is isomorphic to a semidirect product of a group $K^{(m,n)}$, with the Artin group $A(D_n)$, where $K^{(m,n)}$ depends only on the number $m$ of cycles and on the number $n$ of vertices of the graph $T$.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p18Matching and Independence Complexes Related to Small Grids2017-10-19T21:17:01+11:00Benjamin Braunbraunuk@gmail.comWesley K. Houghwesley.hough@uky.edu<p>The topology of the matching complex for the $2\times n$ grid graph is mysterious. We describe a discrete Morse matching for a family of independence complexes $\mathrm{Ind}(\Delta_n^m)$ that include these matching complexes. Using this matching, we determine the dimensions of the chain spaces for the resulting Morse complexes and derive bounds on the location of non-trivial homology groups for certain $\mathrm{Ind}(\Delta_n^m)$. Further, we determine the Euler characteristic of $\mathrm{Ind}(\Delta_n^m)$ and prove that several homology groups of $\mathrm{Ind}(\Delta_n^m)$ are non-zero.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p19Perfect Matchings of Trimmed Aztec Rectangles2017-10-19T21:18:24+11:00Tri Laitlai3@unl.eduWe consider several new families of subgraphs of the square grid whose matchings are enumerated by powers of several small prime numbers: $2$, $3$, $5$, and $11$. Our graphs are obtained by trimming two opposite corners of an Aztec rectangle. The result yields a proof of a conjecture posed by Ciucu. In addition, we reveal a hidden connection between our graphs and the hexagonal dungeons introduced by Blum.2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p20Linear Chord Diagrams with Long Chords2017-10-19T21:19:14+11:00Everett Sullivaneverett.n.sullivan.gr@dartmouth.eduA linear chord diagram of size $n$ is a partition of the set $\{1,2,\dots,2n\}$ into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the effect starts.2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p21Large Cuts with Local Algorithms on Triangle-Free Graphs2017-10-19T21:25:18+11:00Juho Hirvonenjuho.hirvonen@aalto.fiJoel Rybickijoel.rybicki@helsinki.fiStefan Schmidschmiste@cs.aau.dkJukka Suomelajukka.suomela@aalto.fiLet $G$ be a $d$-regular triangle-free graph with $m$ edges. We present an algorithm which finds a cut in $G$ with at least $(1/2 + 0.28125/\sqrt{d})m$ edges in expectation, improving upon Shearer's classic result. In particular, this implies that any $d$-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of $G$.<br /><br />Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient <em>randomised distributed (local) algorithm</em>: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using <em>computational techniques</em>. We show that for any fixed $d$, there exists a weighted neighbourhood graph $\mathcal{N}_d$ such that there is a one-to-one correspondence between heavy cuts of $\mathcal{N}_d$ and randomised local algorithms that find large cuts in any $d$-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in $d$-regular graphs: we can compute the optimal cut of $\mathcal{N}_d$ to attain a lower bound on the maximum cut size of any $d$-regular triangle-free graph.<br /><br />2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p22Maximum Size of a Family of Pairwise Graph-Different Permutations2017-10-19T21:20:45+11:00Louis Golowichlouis.golowich@gmail.comChiheon Kimchiheonk@math.mit.eduRichard Zhourichard.c.zhou@gmail.com<p>Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p23Cover $k$-Uniform Hypergraphs by Monochromatic Loose Paths2017-10-19T21:26:05+11:00Changhong Luchlu@math.ecnu.edu.cnRui Maomaorui1111@163.comBing Wangwuyuwuyou@126.comPing Zhangmathzhangping@126.com<p>A conjecture of Gyárfás and Sárközy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $\mathcal{K}_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k-2$ vertices. Recently, the authors affirmed the conjecture. In the note we show that for every $2$-coloring of $\mathcal{K}_n^k$, one can find two monochromatic paths of distinct colors to cover all vertices of $\mathcal{K}_n^k$ such that they share at most $k-2$ vertices. Omidi and Shahsiah conjectured that $R(\mathcal{P}_t^k,\mathcal{P}_t^k) =t(k-1)+\lfloor \frac{t+1}{2}\rfloor$ holds for $k\ge 3$ and they affirmed the conjecture for $k=3$ or $k\ge 8$. We show that if the conjecture is true, then $k-2$ is best possible for our result.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p24Induced Ramsey-Type Results and Binary Predicates for Point Sets2017-10-19T21:27:24+11:00Martin Balkobalko@kam.mff.cuni.czJan Kynčlkyncl@kam.mff.cuni.czStefan Langermanstefan.langerman@ulb.ac.beAlexander Pilzalexander.pilz@inf.ethz.ch<p>Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Nešetřil and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey.</p><p>As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Nešetřil and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p25Orbits of Antichains in Certain Root Posets2017-10-19T21:31:00+11:00Chao-Ping Dongchaoping@hnu.edu.cnSuijie Wangwangsuijie@hnu.edu.cnBuilding everything from scratch, we give another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$ equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be interpreted as sub-families of certain root posets. We guess these root posets should provide a unified setting to exhibit the homomesy phenomenon defined by Propp and Roby.2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p26Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability2017-10-19T21:31:48+11:00Thomas Mahoneytmahoney@emporia.edu<p>Let $a,b\in\mathbb{N}$. A graph $G$ is $(a,b)$-choosable if for any list assignment $L$ such that $|L(v)|\ge a$, there exists a coloring in which each vertex $v$ receives a set $C(v)$ of $b$ colors such that $C(v)\subseteq L(v)$ and $C(u)\cap C(w)=\emptyset$ for any $uw\in E(G)$. In the online version of this problem, on each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of these vertices to receive that color. We say $G$ is $(a,b)$-paintable if when each vertex $v$ is allowed to be marked $a$ times, there is an algorithm to produce a coloring in which each vertex $v$ receives $b$ colors such that adjacent vertices receive disjoint sets of colors.</p><p>We show that every odd cycle $C_{2k+1}$ is $(a,b)$-paintable exactly when it is $(a,b)$-chosable, which is when $a\ge2b+\lceil b/k\rceil$. In 2009, Zhu conjectured that if $G$ is $(a,1)$-paintable, then $G$ is $(am,m)$-paintable for any $m\in\mathbb{N}$. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any $m \in \mathbb{N}$ that $G$ is $(5m,m)$-paintable when $G$ is planar. Strengthening work of Tuza and Voigt, and of Hladky, Kral, and Schauz, we prove that for any connected graph $G$ other than an odd cycle or complete graph and any $m\in\mathbb{N}$, $G$ is $(\Delta(G)m,m)$-paintable.</p>2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p27Eulerian Polynomials, Stirling Permutations of the Second Kind and Perfect Matchings2017-10-19T21:32:10+11:00Shi-Mei Mashimeimapapers@163.comYeong-Nan Yehmayeh@math.sinica.edu.twIn this paper, we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles. Moreover, we get an expansion of the ordinary derangement polynomials in terms of the Stirling derangement polynomials. Finally, we present constructive proofs of a kind of combinatorial expansions of the Eulerian polynomials of types $A$ and $B$.2017-10-20T00:00:00+11:00Copyright (c) 2017 The Electronic Journal of Combinatoricshttp://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p28A Step Towards Yuzvinsky's Conjecture2017-11-03T12:11:28+11:00Isidoro Gitlerigitler@math.cinvestav.edu.mxEnrique Reyesereyes@math.cinvestav.mxFrancisco Javier Zaragoza Martínezfranz@correo.azc.uam.mx<pre>An intercalate matrix <span>$M$</span> of type <span>$[r,s,n]$</span> is an <span>$r\times s$</span> matrix with entries in <span>$\{1,2,\dotsc,n\}$</span> such that all entries in each row are distinct, all entries in each column are distinct, and all <span>$2 \times 2$</span> submatrices of <span>$M$</span> have either <span>$2$</span> or <span>$4$</span> distinct entries. Yuzvinsky's Conjecture on intercalate matrices claims that the smallest <span>$n$</span> for which there is an intercalate matrix of type <span>$[r,s,n]$</span> is the Hopf-Stiefel function <span>$r \circ s$</span>. In this paper we prove that Yuzvinsky's Conjecture is asimptotically true for <span>$\frac{5}{6}$</span> of integer pairs <span>$(r,s)$</span>. We prove the Conjecture for <span>$r\le 8$,</span> and we study it in the range <span>$r,s\le 32$</span>.<!--EndFragment--></pre>2017-11-03T00:00:00+11:00Copyright (c) 2017 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p29Neighborhood Growth Dynamics on the Hamming Plane2017-11-03T12:11:36+11:00Janko Gravnergravner@math.ucdavis.eduDavid Sivakoffdsivakoff@stat.osu.eduErik Slivkeneslivken@math.univ-paris-diderot.frWe initiate the study of general neighborhood growth dynamics on two-dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.2017-11-03T00:00:00+11:00Copyright (c) 2017 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p30Avoidability of Formulas with Two Variables2017-11-03T12:11:40+11:00Pascal Ochemochem@lirmm.frMatthieu Rosenfeldmatthieu.rosenfeld@ens-lyon.frIn combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words.<br /><br />2017-11-03T00:00:00+11:00Copyright (c) 2017 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p31Antimagic Orientation of Biregular Bipartite Graphs2017-11-03T12:11:45+11:00Songling Shansongling.shan@vanderbilt.eduXiaowei Yuxwyu2013@163.com<p>An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph $G$ is said to have an antimagic orientation if $G$ has an orientation which admits an antimagic labeling. Hefetz, Mütze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that every biregular bipartite graph admits an antimagic orientation.</p>2017-11-03T00:00:00+11:00Copyright (c) 2017 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i4p32A Note on Chromatic Number and Induced Odd Cycles2017-11-03T12:11:51+11:00Baogang Xubaogxu@njnu.edu.cnGexin Yugyu@wm.eduXiaoya Zhaxzha@mtsu.eduAn odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph $G$ has no odd holes then $\chi(G)\le 2^{2^{\omega(G)+2}}$. Chudnovsky, Robertson, Seymour and Thomas showed that if $G$ has neither $K_4$ nor odd holes then $\chi(G)\le 4$. In this note, we show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then $\chi(G)\le 4$ and $\chi(G)\le 3$ if $G$ has radius at most $3$, and for each vertex $u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions in Plummer and Zha (2014).2017-11-03T00:00:00+11:00Copyright (c) 2017