http://www.combinatorics.org/ojs/index.php/eljc/issue/feedThe Electronic Journal of Combinatorics2014-08-21T10:24:23+10:00André Kündgenakundgen@csusm.eduOpen Journal Systems<p>The copyright of published papers remains with the authors. We only require your agreement that we publish it, as described in the following publication release agreement:</p><ol><li>This is an agreement between the Electronic Journal of Combinatorics (the "Journal"), and the copyright owner (the "Owner") of a work (the "Work") to be published in the Journal.</li><li>The Owner warrants that s/he has the full power and authority to enter into this Agreement and to grant the rights granted in this Agreement.</li><li>The Owner hereby grants to the Journal a worldwide, irrevocable, royalty free license to publish or distribute the Work, to enter into arrangements with others to publish or distribute the Work, and to archive the Work.</li><li>The Owner agrees that further publication of the Work, with the same or substantially the same content as appears in the Journal, will include an acknowledgement of prior publication in the Journal.</li></ol><p>The Electronic Journal of Combinatorics (E-JC) is a fully-refereed electronic journal with <em>very high standards,</em> publishing papers of substantial content and interest in all branches of discrete mathematics, including combinatorics, graph theory, and algorithms for combinatorial problems. The journal is <em>completely free</em> for both authors and readers. Authors retain the copyright of their articles. Articles published in E-JC are reviewed on MathSciNet and ZBMath, and are indexed by Web of Science. The latest papers are always available by clicking on the tab marked "Current" near the top of the page. You can also locate papers using the search facility on the right hand side of this page.</p><p>E-JC was founded in 1994 by Herbert S. Wilf and Neil Calkin, making it one of the oldest electronic journals.</p>http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p1A Spectral Equivalent Condition of the $P$-Polynomial Property for Association Schemes2014-08-21T10:24:22+10:00Hiroshi Nozakihnozaki@auecc.aichi-edu.ac.jpHirotake Kuriharakurihara@kct.ac.jpWe give two equivalent conditions of the $P$-polynomial property of a symmetric association scheme. The first equivalent condition shows that the $P$-polynomial property is determined only by the first and second eigenmatrices of the symmetric association scheme. The second equivalent condition is another expression of the first using predistance polynomials.2014-07-03T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p2Consecutive Up-Down Patterns in Up-Down Permutations2014-08-21T10:24:22+10:00Jeffrey B. Remmeljremmel@ucsd.edu<p>In this paper, we study the distribution of the number of consecutive pattern matches of the five up-down permutations of length four, $1324$, $2314$, $2413$, $1432$, and $3412$, in the set of up-down permutations. We show that for any such $\tau$, the generating function for the distribution of the number of consecutive pattern matches of $\tau$ in the set of up-down permutations can be expressed in terms of what we call the generalized maximum packing polynomials of $\tau$. We then provide some systematic methods to compute the generalized maximum packing polynomials for such $\tau$.</p>2014-07-03T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p3A Characteristic Factor for the 3-Term IP Roth Theorem in $\mathbb{Z}_3^\mathbb{N}$2014-08-21T10:24:22+10:00Randall McCutcheonrmcctchn@memphis.eduAlistair Windsorawindsor@memphis.edu<p>Let $\Omega = \bigoplus_{i=1}^\infty \mathbb{Z}_3$ and $e_i = (0, \dots, 0 , 1, 0, \dots)$ where the $1$ occurs in the $i$-th coordinate. Let $\mathscr{F}=\{ \alpha \subset \mathbb{N} : \varnothing \neq \alpha, \alpha \text{ is finite} \}$. <span style="font-size: 10px;">There is a natural inclusion of $\mathscr{F}$ into $\Omega$ where $\alpha \in \mathscr{F}$ is mapped to $e_\alpha = \sum_{i \in \alpha} e_i$. We give a new proof that if $E \subset \Omega$ with $d^*(E) >0$ then there exist $\omega \in \Omega$ and $\alpha \in \mathscr{F}$ such that \[ </span><span style="font-size: 10px;">\{ \omega, \omega+ e_\alpha, \omega + 2 e_\alpha \} \subset E.\]</span><span style="font-size: 10px;">Our proof establishes that for the ergodic reformulation of the problem there is a characteristic factor that is a one step compact extension of the Kronecker factor.</span></p>2014-07-03T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p4On Fence Patrolling by Mobile Agents2014-08-21T10:24:22+10:00Adrian Dumitrescudumitres@uwm.eduAnirban Ghoshanirban@uwm.eduCsaba D. Tóthcdtoth@acm.orgSuppose that a fence needs to be protected (perpetually) by $k$ mobile agents with maximum speeds $v_1,\ldots,v_k$ so that no point on the fence is left unattended for more than a given amount of time. The problem is to determine if this requirement can be met, and if so, to design a suitable patrolling schedule for the agents. Alternatively, one would like to find a schedule that minimizes the <em>idle time</em>, that is, the longest time interval during which some point is not visited by any agent. We revisit this problem, introduced by Czyzowicz et al. (2011), and discuss several strategies for the cases where the fence is an open and a closed curve, respectively.<br /><br />In particular: (i) we disprove a conjecture by Czyzowicz et al. regarding the optimality of their algorithm ${\mathcal A}_2$ for unidirectional patrolling of a closed fence; (ii) we present a schedule with a lower idle time for patrolling an open fence, improving an earlier result of Kawamura and Kobayashi.<br /><br /><br /><br />2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p5Arc-Transitive Dihedral Regular Covers of Cubic Graphs2014-08-21T10:24:22+10:00Jicheng Mama_jicheng@hotmail.com<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>A regular covering projection is called </span><span><em>dihedral</em> </span><span>or </span><span><em>abelian</em> </span><span>if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particular, we classify all arc-transitive dihedral regular covers of $</span><span>K_</span><span>4$</span><span>, $</span><span>K_{</span><span>3</span><span>,</span><span>3}$</span><span>, the 3-cube $</span><span>Q_</span><span>3$ </span><span>and the Petersen graph.</span></p></div></div></div>2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p6Nordhaus-Gaddum Type Inequalities for Laplacian and Signless Laplacian Eigenvalues2014-08-21T10:24:22+10:00F. Ashraffirouzeh_ashraf@yahoo.comB. Tayfeh-Rezaietayfeh-r@ipm.ir<p>Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $\mu_1(G)\ge\cdots\ge\mu_{n-1}(G)\ge\mu_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $\mu_1(G)-\mu_{n-1}(G)\le n-1$ or equivalently $\mu_1(G)+\mu_1(\overline G)\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $\mu_1(G)\mu_1(\overline G)\le n(n-1)$. Aouchiche and Hansen [<em>Discrete Appl. Math.</em> 2013] conjectured that $q_1(G)+q_1(\overline G)\le3n-4$ and $q_1(G)q_1(\overline G)\le2n(n-2)$. We prove the former and disprove the latter by constructing a family of graphs $H_n$ where $q_1(H_n)q_1(\overline{H_n})$ is about $2.15n^2+O(n)$.</p>2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p7The Minimum Number of Nonnegative Edges in Hypergraphs2014-08-21T10:24:22+10:00Hao Huanghuanghao@math.ias.eduBenny Sudakovbenjamin.sudakov@math.ethz.ch<p>An $r$-uniform $n$-vertex hypergraph $H$ is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of $H$. In this paper we show that for $n>10r^3$, every $r$-uniform $n$-vertex hypergraph with equal codegrees has the MMS property, and the bound on $n$ is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of $n>10k^3$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative $k$-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for $n \ge 4k$ and any weighting on the $1$-dimensional subspaces of $\mathbb{F}_{q}^n$ with nonnegative sum, the number of nonnegative $k$-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the <span>Erdős</span><span>-</span><span>Ko</span><span>-</span><span>Rado </span>theorem on $k$-intersecting families.</p>2014-07-10T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p8On the Cayley Isomorphism Problem for Cayley Objects of Nilpotent Groups of Some Orders2014-08-21T10:24:22+10:00Edward Dobsondobson@math.msstate.edu<p>We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group $G$ whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of $G$ in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that ${\mathbb Z}_q\times{\mathbb Z}_p^2\times{\mathbb Z}_m$ is a CI-group with respect to digraphs, where $q$ and $p$ are primes with $p^2 < q$ and $m$ is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on $q$ and $p$).</p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p9Ascent-Descent Young Diagrams and Pattern Avoidance in Alternating Permutations2014-08-21T10:24:22+10:00Ravi Jagadeesanravi.jagadeesan@gmail.com<p><span style="color: #000000;">We investigate pattern avoidance in alternating permutations and an alternating analogue of Young diagrams. </span><span style="color: #000000;">In particular, using an extension of </span><span style="color: #000000; text-decoration: underline;">Babson</span><span style="color: #000000;"> and West's notion of shape-</span><span style="color: #000000; text-decoration: underline;">Wilf</span><span style="color: #000000;"> equivalence </span><span style="color: #000000;">described in our recent paper (with N. </span><span style="color: #000000; text-decoration: underline;">Gowravaram</span><span style="color: #000000;">), we generalize results of </span><span style="color: #000000; text-decoration: underline;">Backelin</span><span style="color: #000000;">, West, and </span><span style="color: #000000; text-decoration: underline;">Xin </span><span style="color: #000000;">and </span><span style="color: #000000; text-decoration: underline;">Ouchterlony </span><span style="color: #000000;">to alternating permutations. Unlike </span><span style="color: #000000; text-decoration: underline;">Ouchterlony</span><span style="color: #000000;"> and Bóna</span><span style="color: #000000;">'s </span><span style="color: #000000; text-decoration: underline;">bijections</span><span style="color: #000000;">, our </span><span style="color: #000000; text-decoration: underline;">bijections</span><span style="color: #000000;"> are not the restrictions of </span><span style="color: #000000; text-decoration: underline;">Backelin</span><span style="color: #000000;">, West, and </span><span style="color: #000000; text-decoration: underline;">Xin's </span><span style="color: #000000; text-decoration: underline;">bijections</span><span style="color: #000000;"> to alternating permutations. </span><span style="color: #000000;">This paper is the second of a two-paper series presenting the work of </span><span style="color: #000000;"><em>Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux </em>(with N. </span><span style="color: #000000; text-decoration: underline;">Gowravaram</span><span style="color: #000000;">, <a href="http://arxiv.org/abs/1301.6796v1">arXiv:</a></span><a href="http://arxiv.org/abs/1301.6796v1"><span style="color: #000000;">1301.</span><span style="color: #000000; text-decoration: underline;">6796v1</span></a><span style="color: #000000;">). The first</span><span style="color: #000000;"> paper in the series is </span><span style="color: #000000;"><em>Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux</em> (with N. </span><span style="color: #000000; text-decoration: underline;">Gowravaram</span><span style="color: #000000;">, </span><span style="color: #000000;"><em>Electronic Journal of </em></span><em><span style="color: #000000;">Combinatorics </span></em><a href="/ojs/index.php/eljc/article/view/v20i4p17" target="_blank"><span style="color: #000000;">20(4):#</span></a><span style="color: #800000;"><a href="/ojs/index.php/eljc/article/view/v20i4p17" target="_blank">P17</a>,</span><span style="color: #000000;"> 2013).</span><span style="color: #000000;"><br /></span></p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p10The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix2014-08-21T10:24:22+10:00Jacob G. Martinjm733@georgetown.eduE. Rodney Canfielderc@cs.uga.edu<p>A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The <em>random Gram matrix</em> of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.</p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p11A Counterexample to a Question of Hof, Knill and Simon2014-08-21T10:24:22+10:00Sébastien Labbélabbe@liafa.univ-paris-diderot.fr<p>In this article, we give a negative answer to a question of Hof, Knill and Simon (1995) concerning purely morphic sequences obtained from primitive morphism containing an infinite number of palindromes. Their conjecture states that such palindromic sequences arise from substitutions that are in class $\mathcal{P}$. The conjecture was proven for the binary alphabet by B. Tan in 2007. We give here a counterexample for a ternary alphabet.<br /><br /></p>2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p12Counting Results for Thin Butson Matrices2014-08-21T10:24:22+10:00Teo Banicateo.banica@gmail.comA partial Butson matrix is a matrix $H\in M_{M\times N}(\mathbb Z_q)$ having its rows pairwise orthogonal, where $\mathbb Z_q\subset\mathbb C^\times$ is the group of $q$-th roots of unity. We investigate here the counting problem for these matrices in the "thin" regime, where $M=2,3,\ldots$ is small, and where $N\to\infty$ (subject to the condition $N\in p\mathbb N$ when $q=p^k>2$). The proofs are inspired from the de Launey-Levin and Richmond-Shallit counting results.2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p13Trivial Meet and Join within the Lattice of Monotone Triangles2014-08-21T10:24:22+10:00John Engbersjohnengbers@gmail.comAdam Hammettadam.hammett@bethelcollege.eduThe lattice of monotone triangles $(\mathfrak{M}_n,\leq)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum $\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$ ($\tau_{\max}$, resp.). Higher-order error terms are also discussed.2014-07-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p14On the Strong Partition Dimension of Graphs2014-08-21T10:24:22+10:00Ismael González Yeroismael.gonzalez@uca.es<p>We present a new style of metric generator in graphs. Specifically we introduce a metric generator based on a partition of the vertex set of a graph. The sets of the partition will work as the elements which will uniquely determine the position of each single vertex of the graph. A set $W$ of vertices of a connected graph $G$ strongly resolves two different vertices $x,y\notin W$ if either $d_G(x,W)=d_G(x,y)+d_G(y,W)$ or $d_G(y,W)=d_G(y,x)+d_G(x,W)$, where $d_G(x,W)=\min\left\{d(x,w)\;:\;w\in W\right\}$. An ordered vertex partition $\Pi=\left\{U_1,U_2,...,U_k\right\}$ of a graph $G$ is a strong resolving partition for $G$ if every two different vertices of $G$ belonging to the same set of the partition are strongly resolved by some set of $\Pi$. A strong resolving partition of minimum cardinality is called a strong partition basis and its cardinality the strong partition dimension. In this article we introduce the concepts of strong resolving partition and strong partition dimension and we begin with the study of its mathematical properties.</p>2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p15The Weak Order on Pattern-Avoiding Permutations2014-08-21T10:24:22+10:00Brian Drakedrakebr@gvsu.eduThe weak order on the symmetric group is a well-known partial order which is also a lattice. We consider subposets of the weak order consisting of permutations avoiding a single pattern, characterizing the patterns for which the subposet is a lattice. These patterns have only a single small ascent or descent. We prove that all patterns for which the subposet is a sublattice have length at most three.2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p16An Erdős-Ko-Rado Theorem for Permutations with Fixed Number of Cycles2014-08-21T10:24:22+10:00Cheng Yeaw Kumatkcy@nus.edu.sgKok Bin Wongkbwong@um.edu.my<p>Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e.,<br />\[ S_{n,k} = \{\pi \in S_{n}: \pi = c_{1}c_{2} \cdots c_{k}\},\] <br />where $c_1,c_2,\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is $\left [ \begin{matrix}n\\ k \end{matrix}\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\mathcal{A} \subseteq S_{n,k}$ is said to be $t$-<em>cycle-intersecting</em> if any two elements of $\mathcal{A}$ have at least $t$ common cycles. In this paper we show that, given any positive integers $k,t$ with $k\geq t+1$, if $\mathcal{A} \subseteq S_{n,k}$ is $t$-cycle-intersecting and $n\ge n_{0}(k,t)$ where $n_{0}(k,t) = O(k^{t+2})$, then <br />\[ |\mathcal{A}| \le \left [ \begin{matrix}n-t\\ k-t \end{matrix}\right],\]<br />with equality if and only if $\mathcal{A}$ is the stabiliser of $t$ fixed points.</p>2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p17Some Identities involving the Partial Sum of $q$-Binomial Coefficients2014-08-21T10:24:22+10:00Bing Heyuhe001@foxmail.comWe give some identities involving sums of powers of the partial sum of $q$-binomial coefficients, which are $q$-analogues of Hirschhorn's identities [<em>Discrete Math.</em> 159 (1996), 273-278] and Zhang's identity [<em>Discrete Math.</em> 196 (1999), 291-298].2014-07-25T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p18Minimum-Weight Edge Discriminators in Hypergraphs2014-08-21T10:24:22+10:00Bhaswar B. Bhattacharyabhaswar.bhattacharya@gmail.comSayantan Dassayantan@umich.eduShirshendu Gangulysganguly@math.washington.eduIn this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an <em>edge-discriminator</em> on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathscr E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathscr E$. An <em>optimal edge-discriminator</em> on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, with $|\mathscr E|=m$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq m(m+1)/2$, and the equality holds if and only if the elements of $\mathscr E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathscr E)$, it follows from earlier results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathscr E)$, with $|\mathscr E|=m~(\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=m(m+1)/2-1$. This shows that all integer values between $m$ and $m(m+1)/2$ cannot be the weight of an optimal edge-discriminator of a hypergraph, and this raises many other interesting combinatorial questions.<br /><br /><br /><br />2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p19The Maximal Length of a $k$-Separator Permutation2014-08-21T10:24:22+10:00Benjamin Gunbybgunby314@gmail.comA permutation $\sigma\in S_n$ is a $k$-separator if all of its patterns of length $k$ are distinct. Let $F(k)$ denote the maximal length of a $k$-separator. Hegarty (2013) showed that $k+\left\lfloor\sqrt{2k-1}\right\rfloor-1\leq F(k)\leq k+\left\lfloor\sqrt{2k-3}\right\rfloor$, and conjectured that $F(k)=k+\left\lfloor\sqrt{2k-1}\right\rfloor-1$. This paper will strengthen the upper bound to prove the conjecture for all sufficiently large $k$ (in particular, for all $k\geq 320801$).2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p20Grid Minors in Damaged Grids2014-08-21T10:24:22+10:00David Eppsteindavid.eppstein@gmail.com<p>We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.</p>2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p21Extremal Problems for the $p$-Spectral Radius of Graphs2014-08-21T10:24:22+10:00Liying Kanglykang@shu.edu.cnVladimir Nikiforovvnikifrv@memphis.edu<p>The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as</p><p>$$\lambda^{(p)}(G) =\max\{ 2\sum_{\{i,j\}\in E(G)} x_ix_j:x_1,\ldots,x_n\in\mathbb{R}\text{ and }\vert x_{1}\vert ^{p}+\cdots+\vert x_n\vert^{p}=1\} .$$</p><p>The most remarkable feature of $\lambda^{(p)}$ is that it seamlessly joins several other graph parameters, e.g., $\lambda^{(1)}$ is the Lagrangian, $\lambda^{(2) }$ is the spectral radius and $\lambda^{(\infty) }/2$ is the number of edges. This paper presents solutions to some extremal problems about $\lambda^{(p)}$, which are common generalizations of corresponding edge and spectral extremal problems.</p><p>Let $T_{r}\left( n\right) $ be the $r$-partite Turán<strong> </strong>graph of order $n$. Two of the main results in the paper are:</p><p>(I) Let $r\geq2$ and $p>1.$ If $G$ is a $K_{r+1}$-free graph of order $n$, then<br />$$\lambda^{(p)}(G) <\lambda^{(p)}(T_{r}(n)),$$ unless $G=T_{r}(n)$.</p><p>(II) Let $r\geq2$ and $p>1.$ If $G\ $is a graph of order $n,$ with</p><p>$$\lambda^{(p)}(G)>\lambda^{(p)}( T_{r}(n)) ,$$</p><p><br />then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1$, where $c$ is a positive number depending only on $p$ and $r.$</p>2014-08-06T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p22On Saturated $k$-Sperner Systems2014-08-21T10:24:22+10:00Natasha Morrisonmorrison@maths.ox.ac.ukJonathan A. Noelnoel@maths.ox.ac.ukAlex Scottscott@maths.ox.ac.uk<p>Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$<em>-Sperner</em> if it does not contain a chain of length $k+1$ under set inclusion and it is <em>saturated</em> if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$.</p><p>A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an <em>oversaturated</em> $k$<em>-Sperner system</em> if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.</p><pre><!--EndFragment--></pre>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p23On the Number of Colored Birch and Tverberg Partitions2014-08-21T10:24:22+10:00Stephan Hellstephan@hell-wie-dunkel.de<p>In 2009, Blagojević, Matschke, and Ziegler established the first tight colored Tverberg theorem. We develop a colored version of our previous results (2008): Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. Both properties follow from similar results on the number of colored Birch partitions.</p>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p24Bounding Sequence Extremal Functions with Formations2014-08-21T10:24:22+10:00Jesse Genesongeneson@math.mit.eduRohil Prasadprasad01@college.harvard.eduJonathan Tidorjtidor@mit.edu<p>An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has no subsequence isomorphic to $u$. For every sequence $u$ define $\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which there exists $r$ such that there is a subsequence isomorphic to $u$ in every $(r, s)$-formation. We use $\mathit{fw}(u)$ to prove upper bounds on $\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternation with the same formation width as $u$.</p><p>We generalize Nivasch's bounds on $\mathit{Ex}((ab)^{t}, n)$ by showing that $\mathit{fw}((12 \ldots l)^{t})=2t-1$ and $\mathit{Ex}((12\ldots l)^{t}, n) =n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for every $l \geq 2$ and $t\geq 3$, such that $\alpha(n)$ denotes the inverse Ackermann function. Upper bounds on $\mathit{Ex}((12 \ldots l)^{t} , n)$ have been used in other papers to bound the maximum number of edges in $k$-quasiplanar graphs on $n$ vertices with no pair of edges intersecting in more than $O(1)$ points.</p><p>If $u$ is any sequence of the form $a v a v' a$ such that $a$ is a letter, $v$ is a nonempty sequence excluding $a$ with no repeated letters and $v'$ is obtained from $v$ by only moving the first letter of $v$ to another place in $v$, then we show that $\mathit{fw}(u)=4$ and $\mathit{Ex}(u, n) =\Theta(n\alpha(n))$. Furthermore we prove that $\mathit{fw}(abc(acb)^{t})=2t+1$ and $\mathit{Ex}(abc(acb)^{t}, n) = n2^{\frac{1}{(t-1)!}\alpha(n)^{t-1}\pm O(\alpha(n)^{t-2})}$ for every $t\geq 2$.</p>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p25Counting the Palstars2014-08-21T10:24:22+10:00L. Bruce Richmondlbrichmo@uwaterloo.caJeffrey O Shallitshallit@cs.uwaterloo.ca<p>A palstar (after Knuth, Morris, and Pratt) is a concatenation of even-length palindromes. We show that, asymptotically, there are $\Theta(\alpha_k^n)$ palstars of length $2n$ over a $k$-letter alphabet, where $\alpha_k$ is a constant such that $2k-1 < \alpha_k < 2k-{1 \over 2}$. In particular, $\alpha_2\doteq 3.33513193$.</p>2014-08-13T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p26A Combinatorial Proof of the Non-Vanishing of Hankel Determinants of the Thue-Morse Sequence2014-08-21T10:24:22+10:00Yann Bugeaudbugeaud@math.unistra.frGuo-Niu Hanguoniu.han@unistra.fr<p>In 1998, Allouche, Peyrière, Wen and Wen established that the Hankel determinants associated with the Thue-Morse sequence on $\{-1,1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p27Note on the Subgraph Component Polynomial2014-08-21T10:24:22+10:00Yunhua Liao307156168@qq.comYaoping Houyphou@hunnu.edu.cn<p>Tittmann, Averbouch and Makowsky [The enumeration of vertex induced subgraphs with respect to the number of components, <em>European J. Combin.</em> 32 (2011) 954-974] introduced the subgraph component polynomial $Q(G;x,y)$ of a graph $G$, which counts the number of connected components in vertex induced subgraphs. This polynomial encodes a large amount of combinatorial information about the underlying graph, such as the order, the size, and the independence number. We show that several other graph invariants, such as the connectivity and the number of cycles of length four in a regular bipartite graph are also determined by the subgraph component polynomial. Then, we prove that several well-known families of graphs are determined by the polynomial $Q(G;x,y).$ Moreover, we study the distinguishing power and find simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the characteristic polynomial, the matching polynomial and the Tutte polynomial. These are partial answers to three open problems proposed by Tittmann et al.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p28Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations2014-08-21T10:24:22+10:00Noah Golowichnoah_g@verizon.netA linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoićič conjectured that the equation $x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoićič's family of equations indeed have a degree of regularity of $n-1$. We also prove a few extensions of this result.<br /><br />2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p29Additional Constructions to Solve the Generalized Russian Cards Problem using Combinatorial Designs2014-08-21T10:24:23+10:00Colleen M. Swansoncmswnsn@umich.eduDouglas R. Stinsondstinson@uwaterloo.ca<p>In the generalized Russian cards problem, we have a card deck $X$ of $n$ cards and three participants, Alice, Bob, and Cathy, dealt $a$, $b$, and $c$ cards, respectively. Once the cards are dealt, Alice and Bob wish to privately communicate their hands to each other via public announcements, without the advantage of a shared secret or public key infrastructure. Cathy, for her part, should remain ignorant of all but her own cards after Alice and Bob have made their announcements. Notions for Cathy's ignorance in the literature range from Cathy not learning the fate of any individual card with certainty (<em>weak $1$-security</em>) to not gaining any probabilistic advantage in guessing the fate of some set of $\delta$ cards (<em>perfect $\delta$-security</em>). As we demonstrate in this work, the generalized Russian cards problem has close ties to the field of combinatorial designs, on which we rely heavily, particularly for perfect security notions. Our main result establishes an equivalence between perfectly $\delta$-secure strategies and $(c+\delta)$-designs on $n$ points with block size $a$, when announcements are chosen uniformly at random from the set of possible announcements. We also provide construction methods and example solutions, including a construction that yields perfect $1$-security against Cathy when $c=2$. Drawing on our equivalence results, we are able to use a known combinatorial design to construct a strategy with $a=8$, $b=13$, and $c=3$ that is perfectly $2$-secure. Finally, we consider a variant of the problem that yields solutions that are easy to construct and optimal with respect to both the number of announcements and level of security achieved. Moreover, this is the first method obtaining weak $\delta$-security that allows Alice to hold an arbitrary number of cards and Cathy to hold a set of $c = \lfloor \frac{a-\delta}{2} \rfloor$ cards. Alternatively, the construction yields solutions for arbitrary $\delta$, $c$ and any $a \geq \delta + 2c$.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p30Capturing the Drunk Robber on a Graph2014-08-21T10:24:23+10:00Natasha Komarovnatalie.komarov@dartmouth.eduPeter Winklerpeter.winkler@dartmouth.edu<p><!--StartFragment-->We show that the expected time for a smart "cop"' to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on $G$; the cop sees all and moves as she wishes, with the object of "capturing" the robber<span>—</span>that is, occupying the same vertex—in least expected time. We show that the cop succeeds in expected time no more than $n {+} {\rm o}(n)$. Since there are graphs in which capture time is at least $n {-} o(n)$, this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.</p>2014-08-21T00:00:00+10:00http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i3p31On Compact Symmetric Regularizations of Graphs2014-08-21T10:24:23+10:00R. Vandellvandellr@ipfw.eduM. Walshwalshm@ipfw.eduW. D. Weakleyweakley@ipfw.edu<p>Let $G$ be a finite simple graph of order $n$, maximum degree $\Delta$, and minimum degree $\delta$. A <em>compact regularization</em> of $G$ is a $\Delta$-regular graph $H$ of which $G$ is an induced subgraph: $H$ is <em>symmetric</em> if every automorphism of $G$ can be extended to an automorphism of $H$. The <em>index</em> $|H:G|$ of a regularization $H$ of $G$ is the ratio $|V(H)|/|V(G)|$. Let $\mbox{mcr}(G)$ denote the index of a minimum compact regularization of $G$ and let $\mbox{mcsr}(G)$ denote the index of a minimum compact symmetric regularization of $G$.</p><p>Erdős and Kelly proved that every graph $G$ has a compact regularization and $\mbox{mcr}(G) \leq 2$. Building on a result of <span>König</span>, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and $\mbox{mcsr}(G) \leq 2^{\Delta - \delta}$. Using a partial Cartesian product construction, we improve this to $\mbox{mcsr}(G) \leq \Delta - \delta + 2$ and give examples to show this bound cannot be reduced below $\Delta - \delta + 1$.</p>2014-08-21T00:00:00+10:00