http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2016-07-21T13:39:27+10:00Andre Kundgenakundgen@csusm.eduOpen Journal Systems<p>The copyright of published papers remains with the authors. 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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/v23i3p14-Factor-Criticality of Vertex-Transitive Graphs2016-07-21T13:39:26+10:00Wuyang Sunswywuyang@163.comHeping Zhangzhanghp@lzu.edu.cn<p>A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical graphs are well-known factor-critical graphs and bicritical graphs, respectively. It is known that if a connected vertex-transitive graph has odd order, then it is factor-critical, otherwise it is elementary bipartite or bicritical. In this paper, we show that a connected vertex-transitive non-bipartite graph of even order at least 6 is 4-factor-critical if and only if its degree is at least 5. This result implies that each connected non-bipartite Cayley graph of even order and degree at least 5 is 2-extendable.</p>2016-07-08T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p2Ramsey Numbers of Trees Versus Odd Cycles2016-07-21T13:39:26+10:00Matthew Brennanbrennanm@mit.eduBurr, Erd<span>ő</span>s, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 25m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.2016-07-08T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p3Chip Games and Paintability2016-07-21T13:39:26+10:00Lech Durajduraj@tcs.uj.edu.plGrzegorz Gutowskigutowski@tcs.uj.edu.plJakub Kozikjkozik@tcs.uj.edu.pl<p>We prove that the paint number of the complete bipartite graph $K_{N,N}$ is $\log N + O(1)$. As a consequence, we get that the difference between the paint number and the choice number of $K_{N,N}$ is $\Theta(\log \log N)$. This answers in the negative the question of Zhu (2009) whether this difference, for all graphs, can be bounded by a common constant. By a classical correspondence, our result translates to the framework of on-line coloring of uniform hypergraphs. This way we obtain that for every on-line two coloring algorithm there exists a $k$-uniform hypergraph with $\Theta(2^k)$ edges on which the strategy fails. The results are derived through an analysis of a natural family of chip games.</p>2016-07-08T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p4A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs2016-07-21T13:39:27+10:00Fan Chungfan@ucsd.edu<p>A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph $G$ with diameter $k$ and vertex set $V$, the smallest nontrivial eigenvalue $\lambda_1$ of the normalized Laplacian $\mathcal L$ satisfies<br />$$<br /> \lambda_1 \leq 1-\sigma \big(1- \frac c {k} \big)<br />$$ for some constant $c$ where $\sigma = 2\sum_v d_v \sqrt{d_v-1}/\sum_v d_v^2 $ and $d_v$ denotes the degree of the vertex $v$.</p><p>We consider weak Ramanujan graphs defined as graphs satisfying $ \lambda_1 \geq 1-\sigma$. We examine the vertex expansion and edge expansion of weak Ramanujan graphs and then use the expansion properties among other methods to derive the above Alon-Boppana bound.</p>2016-07-08T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p5Generalizing the Divisibility Property of Rectangle Domino Tilings2016-07-21T13:39:27+10:00Forest Tongftong@mit.edu<p><!--StartFragment-->We introduce a class of graphs called <em>compound graphs</em>, which are constructed out of copies of a planar bipartite <em>base graph</em>, and explore the number of perfect matchings of compound graphs. The main result is that the number of matchings of every compound graph is divisible by the number of matchings of its base graph. Our approach is to use Kasteleyn's theorem to prove a key lemma, from which the divisibility theorem follows combinatorially. This theorem is then applied to provide a proof of Problem 21 of Propp's <em>Enumeration of Matchings</em>, a divisibility property of rectangles. Finally, we present a new proof, in the same spirit, of Ciucu's factorization theorem.<!--EndFragment--></p>2016-07-08T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p6Orphans in Forests of Linear Fractional Transformations2016-07-21T13:39:27+10:00Sandie Hanshan@citytech.cuny.eduAriane M. Masudaamasuda@citytech.cuny.eduSatyanand Singhssingh@citytech.cuny.eduJohann Thieljthiel@citytech.cuny.edu<p>A positive linear fractional transformation (PLFT) is a function of the form $f(z)=\frac{az+b}{cz+d}$ where $a,b,c$ and $d$ are nonnegative integers with determinant $ad-bc\neq 0$. Nathanson generalized the notion of the Calkin-Wilf tree to PLFTs and used it to partition the set of PLFTs into an infinite forest of rooted trees. The roots of these PLFT Calkin-Wilf trees are called orphans. In this paper, we provide a combinatorial formula for the number of orphans with fixed determinant $D$. In addition, we derive a method for determining the orphan ancestor of a given PLFT. Lastly, taking $z$ to be a complex number, we show that every positive complex number has finitely many ancestors in the forest of complex $(u,v)$-Calkin-Wilf trees.</p>2016-07-08T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p7An $n$-in-a-row Type Game2016-07-21T13:39:27+10:00Joshua Erdejpe282@gmail.comMark Waltersm.walters@qmul.ac.uk<pre>We consider a Maker-Breaker type game on the square grid, in which each</pre><pre> player takes <span>$t$</span> points on their <span>$t^\textrm{th}$</span> turn. Maker wins</pre><pre> if he obtains <span>$n$</span> points on a line (in any direction) without any of</pre><pre> Breaker's points between them. We show that, despite Maker's</pre><pre> apparent advantage, Breaker can prevent Maker from winning until</pre><pre> about his <span>$n^\textrm{th}$</span> turn. We actually prove a stronger</pre><pre> result: Breaker only needs to claim <span>$\omega(\log t)$</span> points on</pre><pre> his <span>$t^\textrm{th}$</span> turn to prevent Maker from winning until this</pre><pre> time. We also consider the situation when the number of points claimed by</pre><pre> Maker grows at other speeds, in particular, when Maker claims</pre><pre> <span>$t^\alpha$</span> points on his <span>$t^\textrm{th}$</span> turn.</pre><pre><!--EndFragment--></pre>2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p8The Topology of the External Activity Complex of a Matroid2016-07-21T13:39:27+10:00Federico Ardilafederico@sfsu.eduFederico Castillofcastillo@ucdavis.eduJosé Alejandro Sampersamper@math.washington.edu<p>We prove that the external activity complex $\textrm{Act}_<(M)$ of a matroid is shellable. In fact, we show that every linear extension of LasVergnas's external/internal order $<_{ext/int}$ on $M$ provides a shelling of $\textrm{Act}_<(M)$. We also show that every linear extension of LasVergnas's internal order $<_{int}$ on $M$ provides a shelling of the independence complex $IN(M)$. As a corollary, $\textrm{Act}_<(M)$ and $M$ have the same $h$-vector. We prove that, after removing its cone points, the external activity complex is contractible if $M$ contains $U_{1,3}$ as a minor, and a sphere otherwise.</p>2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p9Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture2016-07-21T13:39:27+10:00Florent Foucaudflorent.foucaud@gmail.comMichael A. Henningmahenning@uj.ac.zaA total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of $G$, denoted $\gamma_t^L(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \ge 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $\gamma_t^L(G) \le \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $\gamma_t^L(G) \le \frac{3}{4}n$.2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p10Which Haar graphs are Cayley graphs?2016-07-21T13:39:27+10:00István Estélyiestelyii@gmail.comTomaž Pisanskitomaz.pisanski@upr.si<span style="color: #000000;">For a finite group </span><span style="color: #008000;">$G$</span><span style="color: #000000;"> and subset </span><span style="color: #008000;">$S$</span><span style="color: #000000;"> of </span><span style="color: #008000;">$G,$</span><span style="color: #000000;"> the </span><span style="color: #000000;">Haar</span><span style="color: #000000;"> graph </span><span style="color: #008000;">$H(G,S)$</span><span style="color: #000000;"> is a bipartite regular graph, defined as a regular </span><span style="color: #008000;">$G$</span><span style="color: #000000;">-cover of a dipole with </span><span style="color: #008000;">$|S|$ </span><span style="color: #000000;">parallel arcs </span><span style="color: #000000;">labelled</span><span style="color: #000000;"> by elements of </span><span style="color: #008000;">$S$</span><span style="color: #000000;">.</span><span style="color: #000000;"> If </span><span style="color: #008000;">$G$</span><span style="color: #000000;"> is an </span><span style="color: #000000;">abelian</span><span style="color: #000000;"> group, then </span><span style="color: #008000;">$H(G,S)$</span><span style="color: #000000;"> is well-known to be a </span><span style="color: #000000;">Cayley</span><span style="color: #000000;"> graph;</span><span style="color: #000000;"> however, there are examples of non-</span><span style="color: #000000;">abelian</span><span style="color: #000000;"> groups </span><span style="color: #008000;">$G$</span><span style="color: #000000;"> and subsets </span><span style="color: #008000;">$S$</span><span style="color: #000000;"> when this is not the case. In this paper we address the problem of classifying finite non-</span><span style="color: #000000;">abelian</span><span style="color: #000000;"> groups </span><span style="color: #008000;">$G$ </span><span style="color: #000000;">with the property that every </span><span style="color: #000000;">Haar</span><span style="color: #000000;"> graph </span><span style="color: #008000;">$H(G,S)$</span><span style="color: #000000;"> is a </span><span style="color: #000000;">Cayley</span><span style="color: #000000;"> graph. An equivalent condition for </span><span style="color: #008000;">$H(G,S)$</span><span style="color: #000000;"> to be a </span><span style="color: #000000;">Cayley</span><span style="color: #000000;"> graph of a group containing </span><span style="color: #008000;">$G$</span><span style="color: #000000;"> is derived</span><span style="color: #000000;"> in terms of </span><span style="color: #008000;">$G, S$</span><span style="color: #000000;"> and </span><span style="color: #008000;">$\mathrm{Aut } G$</span><span style="color: #000000;">. It is also shown that the </span><pre style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;"><span style="color: #000000;">dihedral</span><span style="color: #000000;"> groups, which are solutions to the above problem, are </span><span style="color: #008000;">$\mathbb{Z}_2^2,D_3,D_4$</span><span style="color: #000000;"> and </span></pre><pre style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;"><span style="color: #008000;">$D_{5}$</span><span style="color: #000000;">.</span></pre><pre style="-qt-paragraph-type: empty; -qt-block-indent: 0; text-indent: 0px; margin: 0px;"> </pre>2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p11Szemerédi's Regularity Lemma via Martingales2016-07-21T13:39:27+10:00Pandelis Dodospdodos@math.uoa.grVassilis Kanellopoulosbkanel@math.ntua.grThodoris Karageorgostkarageo@math.uoa.grWe prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in $L_p$ for any $p>1$. Our approach is based on martingale difference sequences.2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p12The Černý Conjecture and 1-Contracting Automata2016-07-21T13:39:27+10:00Henk Donhenkdon@gmail.com<p>A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. <span>Černý </span>conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. We introduce the notion of aperiodically 1-contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the <span>Černý</span> conjecture holds for aperiodically 1-contracting automata. As a special case, we prove some results for circular automata.</p>2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p13Higher Bruhat Orders in Type B2016-07-21T13:39:27+10:00Seth Shelley-Abrahamsonsethsa@mit.eduSuhas Vijaykumarsuhasv@mit.eduMotivated by the geometry of hyperplane arrangements, Manin and Schechtman defined for each integer $n \geq 1$ a hierarchy of finite partially ordered sets $B(n, k),$ indexed by positive integers $k$, called the higher Bruhat orders. The poset $B(n, 1)$ is naturally identified with the weak left Bruhat order on the symmetric group $S_n$, each $B(n, k)$ has a unique maximal and a unique minimal element, and the poset $B(n, k + 1)$ can be constructed from the set of maximal chains in $B(n, k)$. Ben Elias has demonstrated a striking connection between the posets $B(n, k)$ for $k = 2$ and the diagrammatics of Bott-Samelson bimodules in type A, providing significant motivation for the development of an analogous theory of higher Bruhat orders in other Cartan-Killing types, particularly for $k = 2$. In this paper we present a partial generalization to type B, complete up to $k = 2$, prove a direct analogue of the main theorem of Manin and Schechtman, and relate our construction to the weak Bruhat order and reduced expression graph for Weyl group $B_n$.2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p14Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions2016-07-21T13:39:27+10:00Pavel Galashingalashin@mit.eduDarij Grinbergdarijgrinberg@gmail.comGaku Liugakuliu@math.mit.eduThe dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.2016-07-22T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p15Proof of Gessel's $\gamma$-Positivity Conjecture2016-07-21T13:39:27+10:00Zhicong Linlin@nims.re.krWe prove a conjecture of Gessel, which asserts that the joint distribution of descents and inverse descents on permutations has a fascinating refined $\gamma$-positivity.2016-07-22T00:00:00+10:00