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Showing papers in "Electronic Journal of Combinatorics in 2014"


Journal ArticleDOI
TL;DR: In this article, the Laplacian tensor tensor of a regular hypergraph is derived from the spectrum of the degree sequence of the hypergraph, and the spectral properties of power hypergraphs are studied.
Abstract: For a $k$-uniform hypergraph $H$, we obtain some trace formulas for the Laplacian tensor of $H$, which imply that $\sum_{i=1}^nd_i^s$ ($s=1,\ldots,k$) is determined by the Laplacian spectrum of $H$, where $d_1,\ldots,d_n$ is the degree sequence of $H$. Using trace formulas for the Laplacian tensor, we obtain expressions for some coefficients of the Laplacian polynomial of a regular hypergraph. We give some spectral characterizations of odd-bipartite hypergraphs, and give a partial answer to a question posed by Shao et al (2014). We also give some spectral properties of power hypergraphs, and show that a conjecture posed by Hu et al (2013) holds under certain conditons.

62 citations


Journal ArticleDOI
TL;DR: In this article, combinatorial invariants related to the integer decomposition property of dilated polytopes are proposed and studied, and a fundamental question is to determine the integers for which the polytope possesses the integer decomposition property.
Abstract: An integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ possesses the integer decomposition property if, for any integer $k > 0$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^{N}$, there exist $\alpha_{1}, \ldots, \alpha_k \in \mathcal{P} \cap \mathbb{Z}^{N}$ such that $\alpha = \alpha_{1} + \cdots + \alpha_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

39 citations


Journal ArticleDOI
TL;DR: Bondarenko's 65-dimensional counterexample to Borsuk's conjecture contains a 64-dimensional set of 352 points that is a two-distance set of 351 points.
Abstract: Bondarenko's 65-dimensional counterexample to Borsuk's conjecture contains a 64-dimensional counterexample. It is a two-distance set of 352 points.

29 citations


Journal ArticleDOI
TL;DR: It is proved that if G is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition, the first instance of such a theorem, in which the tree is different from a path or a star.
Abstract: We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar a t and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013). Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant.

28 citations


Journal ArticleDOI
TL;DR: It is proved that a set of lines in $\mathsf{PG}(d,\mathbb{F})$ meeting each hyperplane in a generator set of points has to contain at least $\lfloor1.5d\rfloor$ lines, proving that the lower bound is tight over algebraically closed fields.
Abstract: In this article, we examine sets of lines in $\mathsf{PG}(d,\mathbb{F})$ meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least $\lfloor1.5d\rfloor$ lines if the field $\mathbb{F}$ has at least $\lfloor1.5d\rfloor$ elements, and at least $2d-1$ lines if the field $\mathbb{F}$ is algebraically closed. We show that suitable $2d-1$ lines constitute such a set (if $|\mathbb{F}|\ge2d-1$), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong $(s,A)$ subspace designs constructed by Guruswami and Kopparty have better (smaller) parameter $A$ than one would think at first sight.

25 citations


Journal ArticleDOI
TL;DR: It is proved that the list-chromatic index and paintability index of K_{p+1} is p, for all odd primes, which implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices.
Abstract: We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensatze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.

25 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that for strong amalgamation classes of finite structures with disjoint finite signatures, the class of all finite structures whose π-reductions are from π and τ-redUCTions from the same signature is also Ramsey.
Abstract: Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be strong amalgamation classes of finite structures, with disjoint finite signatures $\sigma$ and $\tau$. Then $\mathcal{C}_1 \wedge \mathcal{C}_2$ denotes the class of all finite ($\sigma\cup\tau$)-structures whose $\sigma$-reduct is from $\mathcal{C}_1$ and whose $\tau$-reduct is from $\mathcal{C}_2$. We prove that when $\mathcal{C}_1$ and $\mathcal{C}_2$ are Ramsey, then $\mathcal{C}_1 \wedge \mathcal{C}_2$ is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.

24 citations


Journal ArticleDOI
Kevin Woods1
TL;DR: Chen, Li, Sam, Calegari, Walker, and Roune, Woods as mentioned in this paper conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets $S_t\subseteq\mathbb{N}^d$ that are defined with quantifiers and statements of the form $a_1(t)x_1+\cdots+a_dx_d\le b(t).
Abstract: A function $g$, with domain the natural numbers, is a quasi-polynomial if there exists a period $m$ and polynomials $p_0,p_1,\ldots,p_{m-1}$ such that $g(t)=p_i(t)$ for $t\equiv i\bmod m$. Quasi-polynomials classically - and "reasonably'' - appear in Ehrhart theory and in other contexts where one examines a family of polyhedra, parametrized by a variable $t$, and defined by linear inequalities of the form $a_1x_1+\cdots+a_dx_d\le b(t)$. Recent results of Chen, Li, Sam; Calegari, Walker; and Roune, Woods show a quasi-polynomial structure in several problems where the $a_i$ are also allowed to vary with $t$. We discuss these "unreasonable'' results and conjecture a general class of sets that exhibit various (eventual) quasi-polynomial behaviors: sets $S_t\subseteq\mathbb{N}^d$ that are defined with quantifiers ($\forall$, $\exists$), boolean operations (and, or, not), and statements of the form $a_1(t)x_1+\cdots+a_d(t)x_d \le b(t)$, where $a_i(t)$ and $b(t)$ are polynomials in $t$. These sets are a generalization of sets defined in the Presburger arithmetic. We prove several relationships between our conjectures, and we prove several special cases of the conjectures. The title is a play on Eugene Wigner's "The unreasonable effectiveness of mathematics in the natural sciences''.

23 citations


Journal ArticleDOI
TL;DR: In this article, Czyzowicz et al. presented a schedule with a lower idle time for patrolling an open fence, improving an earlier result of Kawamura and Kobayashi, and discussed several strategies for the cases where the fence is an open and a closed curve.
Abstract: Suppose 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 idle time , 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. 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.

23 citations


Journal ArticleDOI
TL;DR: In this article, the authors provide an explicit Dynkin diagrammatic description of the $c$ -vectors and the $d$-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix.
Abstract: We provide an explicit Dynkin diagrammatic description of the $c$-vectors and the $d$-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types $A_n$ and $D_n$, then we apply the folding method to $D_{n+1}$ and $A_{2n-1}$ to obtain types $B_n$ and $C_n$. Exceptional types are done by direct inspection with the help of a computer algebra software. We also propose a conjecture on the root property of $c$-vectors for a general cluster algebra.

23 citations


Journal ArticleDOI
TL;DR: The main purpose of this paper is to derive various Matiyasevich-Miki-Gessel type convolution identities for Bernoulli and Genocchi polynomials and numbers by applying some Euler type identities with two parameters.
Abstract: The main purpose of this paper is to derive various Matiyasevich-Miki-Gessel type convolution identities for Bernoulli and Genocchi polynomials and numbers by applying some Euler type identities with two parameters.

Journal ArticleDOI
TL;DR: This work studies the symmetry behavior of preferential and uniform attachment graphs, and introduces a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and it is shown that the change introduces symmetry for all values of $m$.
Abstract: Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.

Journal ArticleDOI
Tri Lai1
TL;DR: In this paper, the authors generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered aztec rectangles.
Abstract: We generalize a theorem of W. Jockusch and J. Propp on quartered Aztec diamonds by enumerating the tilings of quartered Aztec rectangles. We use subgraph replacement method to transform the dual graph of a quartered Aztec rectangle to the dual graph of a quartered lozenge hexagon, and then use Lindstr o m-Gessel-Viennot methodology to find the number of tilings of a quartered lozenge hexagon.

Journal ArticleDOI
TL;DR: A general lower bound is given for the number of homomorphisms from a tree to any graph and it is shown that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the stargraph has the most.
Abstract: In this paper we study several problems concerning the number of homomorphisms of trees We begin with an algorithm for the number of homomorphisms from a tree to any graph By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees These applications include a far reaching generalization and a dual of Bollobas and Tyomkyn's result concerning the number of walks in trees Some other main results of the paper are the following Denote by $\hom(H,G)$ the number of homomorphisms from a graph $H$ to a graph $G$ For any tree $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$ As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$ We also show that if $T_m$ is any fixed tree and $$\hom(T_m,P_n)>\hom(T_m,T_n),$$for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$ All the results together enable us to show that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the star graph has the most

Journal ArticleDOI
TL;DR: In this article, the authors established a relationship between the total transversal number and the total domination number of uniform hypergraphs, and proved tight asymptotic upper bounds on the number of vertices, edges, and edges.
Abstract: In 2012, the first three authors established a relationship between the transversal number and the domination number of uniform hypergraphs. In this paper, we establish a relationship between the total transversal number and the total domination number of uniform hypergraphs. We prove tight asymptotic upper bounds on the total transversal number in terms of the number of vertices, the number of edges, and the edge size.

Journal ArticleDOI
TL;DR: In this article, the maximum possible girth of a regular hypergraph with at most n vertices has been shown to be bounded by a factor of between $3/2+o(1)$ and $2 +o( 1)$ with high probability.
Abstract: We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between $3/2+o(1)$ and $2 +o(1)$). We also define a random $r$-uniform 'Cayley' hypergraph on the symmetric group $S_n$ which has girth $\Omega (\sqrt{\log |S_n|})$ with high probability, in contrast to random regular $r$-uniform hypergraphs, which have constant girth with positive probability.

Journal ArticleDOI
TL;DR: A theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph is extended toKneser hypergraphs and a lower bound for the local chromatic number is derived.
Abstract: Using a $\mathbb{Z}_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a uniform hypergraph).

Journal ArticleDOI
TL;DR: In this paper, the authors derived explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions of the Young diagram, including hooks and the $2\times 2$ box, and provided an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants.
Abstract: We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_{\lambda}$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $\lambda$, including hooks and the $2\times 2$ box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $\lambda$ is a hook plus a box at the $(2,2)$ corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov. This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.

Journal ArticleDOI
TL;DR: It is proved that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph.
Abstract: Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph. As a consequence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the influence of “cycle parity” on the growth rate, compare the growth rates of geometric grid classes against those of the corresponding monotone grid classes, and present new results concerning the effect of edge subdivision on the largest root of the matching polynomial.

Journal ArticleDOI
TL;DR: The exponential generating function for permutations with all valleys even and all peaks odd is found, and it is used to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu.
Abstract: We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers: \[ \left(1-E_1x+E_{3}\frac{x^{3}}{3!}-E_{4}\frac{x^{4}}{4!}+E_{6}\frac{x^{6}}{6!}-E_{7}\frac{x^{7}}{7!}+\cdots\right)^{-1}, \tag{$*$} \] where $\sum_{n=0}^\infty E_n x^n\!/n! = \sec x + \tan x$. We give two proofs of this formula. The first uses a system of differential equations whose solution gives the generating function \begin{equation*} \frac{3\sin\left(\frac{1}{2}x\right)+3\cosh\left(\frac{1}{2}\sqrt{3}x\right)}{3\cos\left(\frac{1}{2}x\right)-\sqrt{3}\sinh\left(\frac{1}{2}\sqrt{3}x\right)}, \end{equation*} which we then show is equal to $(*)$. The second proof derives $(*)$ directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function $(*)$ is an "alternating" analogue of David and Barton's generating function \[ \left(1-x+\frac{x^{3}}{3!}-\frac{x^{4}}{4!}+\frac{x^{6}}{6!}-\frac{x^{7}}{7!}+\cdots\right)^{-1}, \] for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel.

Journal ArticleDOI
TL;DR: The construction of a contracted pseudotriangulation of a 3-manifold $M$ is based on a presentation of the fundamental group of £M and it is computer-free.
Abstract: We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold $M$ is at least the weight of the fundamental group of $M$. This lower bound is sharp for the 3-manifolds $\mathbb{RP}^3$, $L(3,1)$, $L(5,2)$, $S^1\times S^1 \times S^1$, $S^2 \times S^1$, twisted product of $S^2$ and $S^1$ and $S^3/Q_8$, where $Q_8$ is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We have also constructed contracted pseudotriangulations of $L(kq-1,q)$ with $4(q+k-1)$ facets for $q \geq 3$, $k \geq 2$ and $L(kq+1,q)$ with $4(q+k)$ facets for $q\geq 4$, $k\geq 1$. By a recent result of Swartz, our pseudotriangulations of $L(kq+1, q)$ are facet minimal when $kq+1$ are even. In 1979, Gagliardi found presentations of the fundamental group of a manifold $M$ in terms of a contracted pseudotriangulation of $M$. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold $M$, we construct a contracted pseudotriangulation of $M$. So, our construction of a contracted pseudotriangulation of a 3-manifold $M$ is based on a presentation of the fundamental group of $M$ and it is computer-free.

Journal ArticleDOI
TL;DR: This paper characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary it answers an open question about leaving out elements from shatterable set systems in the case of families of Vapnik’s dimension $2.
Abstract: We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=\{F~\cap~S ~:~F~\in~\mathcal{F}\}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the age of any homogeneous directed graph allows a Ramsey precompact expansion, and the relative expansion properties of the universal minimal flow were verified.
Abstract: In 2005, Kechris, Pestov and Todor c evi c provided a powerful tool to compute an invariant of topological groups known as the universal minimal flow, immediately leading to an explicit representation of this invariant in many concrete cases. More recently, the framework was generalized allowing for further applications, and the purpose of this paper is to apply these new methods in the context of homogeneous directed graphs. In this paper, we show that the age of any homogeneous directed graph allows a Ramsey precompact expansion . Moreover, we verify the relative expansion properties and consequently describe the respective universal minimal flows.

Journal ArticleDOI
TL;DR: In this article, it was shown that the M o bius function is unbounded on the poset of all permutations, where the permutations are ordered by pattern containment, and the set of permutations is a poset.
Abstract: The set of all permutations, ordered by pattern containment, is a poset. We give a formula for the M o bius function of intervals $[1,\pi]$ in this poset, for any permutation $\pi$ with at most one descent. We compute the M o bius function as a function of the number and positions of pairs of consecutive letters in $\pi$ that are consecutive in value. As a result of this we show that the M o bius function is unbounded on the poset of all permutations. We show that the M o bius function is zero on any interval $[1,\pi]$ where $\pi$ has a triple of consecutive letters whose values are consecutive and monotone. We also conjecture values of the M o bius function on some other intervals of permutations with at most one descent.

Journal ArticleDOI
TL;DR: Several properties of Turan density, such as supersaturation, blow-up, and suspension, are generalized from uniform hyper graphs to non-uniform hypergraphs and the Turan-densities of $\{1,2\}$-hypergraphs are completely determine.
Abstract: A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turan density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This concept, which generalizes the Turan density of $k$-uniform hypergraphs, is motivated by recent work on extremal poset problems. The details connecting these two areas will be revealed in the end of this paper. Several properties of Turan density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Turan densities of $\{1,2\}$-hypergraphs.

Journal ArticleDOI
TL;DR: In this paper, Mahmoudi et al. showed that the Castelnuovo-Mumford regularity of well-covered graphs is equivalent to vertex-decomposability, codismantlability and Cohen-Macaulayness.
Abstract: We call a vertex $x$ of a graph $G=(V,E)$ a codominated vertex if $N_G[y]\subseteq N_G[x]$ for some vertex $y\in V\backslash \{x\}$, and a graph $G$ is called codismantlable if either it is an edgeless graph or it contains a codominated vertex $x$ such that $G-x$ is codismantlable. We show that $(C_4,C_5)$-free vertex-decomposable graphs are codismantlable, and prove that if $G$ is a $(C_4,C_5,C_7)$-free well-covered graph, then vertex-decomposability, codismantlability and Cohen-Macaulayness for $G$ are all equivalent. These results complement and unify many of the earlier results on bipartite, chordal and very well-covered graphs. We also study the Castelnuovo-Mumford regularity $reg(G)$ of such graphs, and show that $reg(G)=im(G)$ whenever $G$ is a $(C_4,C_5)$-free vertex-decomposable graph, where $im(G)$ is the induced matching number of $G$. Furthermore, we prove that $H$ must be a codismantlable graph if $im(H)=reg(H)=m(H)$, where $m(H)$ is the matching number of $H$. We further describe an operation on digraphs that creates a vertex-decomposable and codismantlable graph from any acyclic digraph. By way of application, we provide an infinite family $H_n$ ($n\geq 4$) of sequentially Cohen-Macaulay graphs whose vertex cover numbers are half of their orders, while containing no vertex of degree-one such that they are vertex-decomposable, and $reg(H_n)=im(H_n)$ if $n\geq 6$. This answers a recent question of Mahmoudi et al.

Journal ArticleDOI
TL;DR: All sets of points of both affine and projective spaces over the Galois field $\mathop{\rm{GF}}(q)$ such that every line of the geometry that is neither contained in £X nor disjoint from £X meets the set $X$ in a constant number of points are determined.
Abstract: In this paper we study sets $X$ of points of both affine and projective spaces over the Galois field $\mathop{\rm{GF}}(q)$ such that every line of the geometry that is neither contained in $X$ nor disjoint from $X$ meets the set $X$ in a constant number of points and we determine all such sets This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [ Designs, Codes and Crypt , 2014] We prove that, up to complements, in $\mathop{\rm{PG}}(n,q)$ such a set $X$ is either a subspace or $n=2,q$ is even and $X$ is a maximal arc of degree $m$ In $\mathop{\rm{AG}}(n,q)$ we show that $X$ is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree $m$ (or the complement of a maximal arc) or a cylinder with base the projection of a quadric Finally we show that in the affine case there are examples (different from subspaces or their complements) in $\mathop{\rm{AG}}(n,4)$ and in $\mathop{\rm{AG}}(n,16)$ giving new neighbour transitive codes in Johnson graphs

Journal ArticleDOI
TL;DR: In this article, it was shown that there is a polynomial lower bound for the minimum size of a saturated k-Sperner system with cardinality at most 2π(1-varepsilon)k.
Abstract: Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$ -Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated 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 $2^{(1-\varepsilon)k}$. A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$ -Sperner system 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.

Journal ArticleDOI
TL;DR: A cyclic analogue of the MFS-action on derangements is defined, and a combinatorial interpretation of the expansion of the derangement polynomial is given on the basis of q(1 + q)^{n-1-2k}, k = 0, 1,..., lfloor ( n-1)/2\rfloor.
Abstract: In this paper we define a cyclic analogue of the MFS-action on derangements, and give a combinatorial interpretation of the expansion of the $n$-th derangement polynomial on the basis $\{q^k(1 + q)^{n-1-2k}, k = 0, 1, , \lfloor (n-1)/2\rfloor \}$

Journal ArticleDOI
TL;DR: The main goal of this paper is to introduce and begin the study of a more general 4-variable polynomial for triangulations and handle decompositions of orientable manifolds.
Abstract: The Tutte polynomial ${T}_G(X,Y)$ of a graph $G$ is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs $G$, $T_G(X,Y) = {T}_{G^*}(Y,X)$ where $G^*$ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial complexes. Polynomial duality for triangulations of a sphere follows as a consequence of Alexander duality. The main goal of this paper is to introduce and begin the study of a more general $4$-variable polynomial for triangulations and handle decompositions of orientable manifolds. Polynomial duality in this case is a consequence of Poincare duality on manifolds. In dimension 2 these invariants specialize to the well-known polynomial invariants of ribbon graphs defined by B. Bollobas and O. Riordan. Examples and specific evaluations of the polynomials are discussed.