http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2017-07-14T11:37:00+10:00Matt Beckmattbeck@sfsu.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><p>ISSN: <span lang="EN">1077-8926</span></p>http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p1On the Nonexistence of $k$-Reptile Simplices in $\mathbb R^3$ and $\mathbb R^4$2017-07-14T11:17:58+10:00Jan Kynčlkyncl@kam.mff.cuni.czZuzana Patákovázuzka@kam.mff.cuni.czA $d$-dimensional simplex $S$ is called a<em> $k$-reptile</em> (or a <em>$k$-reptile simplex</em>) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matoušek and the second author that for $d=3$, $k$-reptile tetrahedra can exist only for $k=m^3$. We then prove a weaker analogue of this result for $d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for $k=m^2$.2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p2$T$-Joins in Infinite Graphs2017-07-14T11:22:41+10:00Attila Joóbalamud21@gmail.com<p>We characterize the class of infinite graphs $G$ for which there exists a $T$-join for any choice of an infinite $T \subseteq V(G)$. We also show that the following well-known fact remains true in the infinite case. If $G$ is connected and does not contain a $T$-join, then it will if we either remove an arbitrary vertex from $T$ or add any new vertex to $T$.</p>2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p3Ideals and Quotients of Diagonally Quasi-Symmetric Functions2017-07-14T11:23:59+10:00Shu Xiao Lilishu3@yorku.caIn 2004, J.-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions $\operatorname{\mathsf{DQSym}}$ in the ring $\mathbb{Q}[\mathbf{x},\mathbf{y}]$ with two sets of variables. They made conjectures on the structure of the quotient $\mathbb{Q}[\mathbf{x},\mathbf{y}]/\langle\operatorname{\mathsf{DQSym}}^+\rangle$, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely many variables i.e. $\mathbf{x}=x_1,x_2,\dots$ and $\mathbf{y}=y_1,y_2,\dots$. Then we apply this construction to the case where there are finitely many variables, and compute the second column of its Hilbert matrix.2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p4The Spectral Gap of Graphs Arising From Substring Reversals2017-07-14T11:25:30+10:00Fan Chungfan@ucsd.eduJosh Tobinrjtobin@ucsd.edu<p>The substring reversal graph $R_n$ is the graph whose vertices are the permutations $S_n$, and where two permutations are adjacent if one is obtained from a substring reversal of the other. We determine the spectral gap of $R_n$, and show that its spectrum contains many integer values. Further we consider a family of graphs that generalize the prefix reversal (or pancake flipping) graph, and show that every graph in this family has adjacency spectral gap equal to one.</p>2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p5Multicolor Ramsey Numbers and Restricted Turán Numbers for the Loose 3-Uniform Path of Length Three2017-07-14T11:27:47+10:00Andrzej Rucińskirucinski@amu.edu.plEliza Jackowskaelijac@amu.edu.plJoanna Polcynjoaska@amu.edu.plLet $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Tur<span>á</span>n number $\mathrm{ex}_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Tur<span>á</span>n numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Tur<span>á</span>n-type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p6Bounds for Distinguishing Invariants of Infinite Graphs2017-07-14T11:29:20+10:00Wilfried Imrichimrich@unileoben.ac.atRafał Kalinowskikalinows@agh.edu.plMonika Pilśniakpilsniak@agh.edu.plMohammad Hadi Shekarrizmh.shekarriz@stu.um.ac.ir<pre>We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.</pre>2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p7Bijection Between Oriented Maps and Weighted Non-Oriented Maps2017-07-14T11:32:17+10:00Agnieszka Czyżewska-Jankowskaagnieszka.czyzewska@gmail.comPiotr Śniadypsniady@impan.pl<p><span>We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.</span></p>2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p8Coloring Graphs with no Even Hole of Length at Least 6: the Triangle-Free Case2017-07-14T11:33:50+10:00Aurélie Lagoutteaurelie.lagoutte@grenoble-inp.fr<pre><!--StartFragment-->In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are <span>$\chi$</span>-bounded but we allow here the existence of <span>$C_4$</span>. The proof relies on the concept of Parity Changing Path, an adaptation of Trinity Changing Path which was recently introduced by <span>Bonamy</span>, <span>Charbit</span> and <span>Thomassé</span> to prove that graphs with no induced cycle of length divisible by three have bounded chromatic number.<!--EndFragment--></pre>2017-07-14T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p9Analysis of the Gift Exchange Problem2017-07-14T11:37:00+10:00Moa Apagodumapagodu@vcu.eduDavid Applegatedavid@bcda.usN. J. A. Sloanenjasloane@gmail.comDoron ZeilbergerDoronZeil@gmail.com<p>In the <em>gift exchange</em> game there are $n$ players and $n$ wrapped gifts. When a player's number is called, that person can either choose one of the remaining wrapped gifts, or can "steal" a gift from someone who has already unwrapped it, subject to the restriction that no gift can be stolen more than a total of $\sigma$ times. The problem is to determine the number of ways that the game can be played out, for given values of $\sigma$ and $n$. Formulas and asymptotic expansions are given for these numbers. This work was inspired in part by a 2005 remark by Robert A. Proctor in the <em>On-Line Encyclopedia of Integer Sequences</em>.</p><p>This is a sequel to the earlier article [arXiv:0907.0513] by the second and third authors, differing from it in that there are two additional authors and several new theorems, including the resolution of most of the conjectures, and the extensive tables have been omitted.</p>2017-07-14T00:00:00+10:00