Showing papers in "Electronic Journal of Combinatorics in 2007"
TL;DR: The perfectness of X_n is decided and it is shown that all nonzero eigenvalues of $X_n$ are integers dividing the value $\varphi(n)$ of the Euler function.
Abstract: The unitary Cayley graph $X_n$ has vertex set $Z_n=\{0,1, \ldots ,n-1\}$. Vertices $a, b$ are adjacent, if gcd$(a-b,n)=1$. For $X_n$ the chromatic number, the clique number, the independence number, the diameter and the vertex connectivity are determined. We decide on the perfectness of $X_n$ and show that all nonzero eigenvalues of $X_n$ are integers dividing the value $\varphi(n)$ of the Euler function.
150 citations
TL;DR: In this article, the notion of quantum chromatic number of a graph was investigated, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.
Abstract: We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is $2$, nor if it is $3$ in a restricted quantum model; on the other hand, we exhibit a graph on $18$ vertices and $44$ edges with chromatic number $5$ and quantum chromatic number $4$.
121 citations
TL;DR: It is proved that the bijection presented is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
Abstract: The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is ${\cal C}_{n} {\cal C}_{n+1}$ where ${\cal C}_{n}={1\over n+1}{2n \choose n}$ is the $n^{th}$ Catalan number. We present a (long awaited) simple bijection which explains this result. Then, we prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
104 citations
TL;DR: The distinguishing number of a graph G is the least cardinal number $D(G)$ such that G has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism.
Abstract: The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$
76 citations
TL;DR: Several interesting formulae for the eigenvalues of the derangement graph are derived and used to settle affirmatively a conjecture of Ku regarding the least eigenvalue.
Abstract: We derive several interesting formulae for the eigenvalues of the derangement graph and use them to settle affirmatively a conjecture of Ku regarding the least eigenvalue.
63 citations
TL;DR: It is shown that this is often the case for random injections, and easy-to-check conditions are provided for the non-trivial task of verifying a negative dependency graph forrandom injections using the Lovasz Local Lemma.
Abstract: The Lovasz Local Lemma is known to have an extension for cases where independence is missing but negative dependencies are under control. We show that this is often the case for random injections, and we provide easy-to-check conditions for the non-trivial task of verifying a negative dependency graph for random injections. As an application, we prove existence results for hypergraph packing and Turan type extremal problems. A more surprising application is that tight asymptotic lower bounds can be obtained for asymptotic enumeration problems using the Lovasz Local Lemma.
62 citations
TL;DR: It is shown that, under mild restrictions, analogous results hold for the alternating group and the direct product of symmetric groups.
Abstract: Let $S_{n}$ denote the symmetric group on $[n]=\{1, \ldots, n\}$. A family $I \subseteq S_{n}$ is intersecting if any two elements of $I$ have at least one common entry. It is known that the only intersecting families of maximal size in $S_{n}$ are the cosets of point stabilizers. We show that, under mild restrictions, analogous results hold for the alternating group and the direct product of symmetric groups.
60 citations
TL;DR: The paper establishes an interesting link between Circuit Complexity and Network Coding, a new direction of research in multiuser information theory.
Abstract: Valiant's shift problem asks whether all $n$ cyclic shifts on $n$ bits can be realized if $n^{(1+\epsilon)}$ input output pairs ($\epsilon As a by-product the paper also establish an interesting link between Circuit Complexity and Network Coding, a new direction of research in multiuser information theory.
57 citations
TL;DR: It is shown that for appropriate choice of the parameters random intersection graphs differ from $G_{n,p}$ in that neither the so-called giant component, appearing when the expected vertex degree gets larger than one, has linear order nor is the second largest of logarithmic order.
Abstract: We study the evolution of the order of the largest component in the random intersection graph model which reflects some clustering properties of real–world networks. We show that for appropriate choice of the parameters random intersection graphs differ from $G_{n,p}$ in that neither the so-called giant component, appearing when the expected vertex degree gets larger than one, has linear order nor is the second largest of logarithmic order. We also describe a test of our result on a protein similarity network.
57 citations
TL;DR: This work uses the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels to prove that every Kneser graph with n-n 6 and k-2 is-distinguishable.
Abstract: This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertex set using $1, \ldots, d$ so that no nontrivial automorphism of $G$ preserves the labels A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$ We prove that a graph is $d$-distinguishable if and only if it has a determining set that can be $(d-1)$-distinguished We use this to prove that every Kneser graph $K_{n:k}$ with $n\geq 6$ and $k\geq 2$ is $2$-distinguishable
56 citations
TL;DR: This note shows that G has a maximum matching which includes all but an exponentially small fraction of the vertices, O((d-1)^{-g/2})$, where the number of unmatched vertices is at most $n/n_0(d,g)$.
Abstract: Let $G=(V,E)$ be any $d$-regular graph with girth $g$ on $n$ vertices, for $d \geq 3$ This note shows that $G$ has a maximum matching which includes all but an exponentially small fraction of the vertices, $O((d-1)^{-g/2})$ Specifically, in a maximum matching of $G$, the number of unmatched vertices is at most $n/n_0(d,g)$, where $n_0(d,g)$ is the number of vertices in a ball of radius $\lfloor(g-1)/2\rfloor$ around a vertex, for odd values of $g$, and around an edge, for even values of $g$ This result is tight if $n
TL;DR: In this article, the authors studied the effect of adding or removing a few edges on the spectral radius of a regular graph and showed that the effect is independent of the number of edges added or removed.
Abstract: Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that $$ \Delta-\lambda_1>{1\over nD}, $$ where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.
TL;DR: It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$.
Abstract: The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number , namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.
TL;DR: The list of all minimal 2-infinite diagrams, which are cluster algebraic analogues of extended Dynkin graphs, is computed.
Abstract: We compute the list of all minimal 2-infinite diagrams, which are cluster algebraic analogues of extended Dynkin graphs.
TL;DR: Some structural problems in $r$-twin-free graphs are studied, such as the existence of the path with $2r+1$ vertices as a subgraph, or the consequences of deleting one vertex.
Abstract: Consider a connected undirected graph $G=(V,E)$, a subset of vertices $C \subseteq V$, and an integer $r \geq 1$; for any vertex $v\in V$, let $B_r(v)$ denote the ball of radius $r$ centered at $v$, i.e., the set of all vertices linked to $v$ by a path of at most $r$ edges. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and different, then we call $C$ an $r$-identifying code. A graph admits at least one $r$-identifying code if and only if it is $r$-twin-free, that is, the sets $B_r(v)$, $v \in V$, are all different. We study some structural problems in $r$-twin-free graphs, such as the existence of the path with $2r+1$ vertices as a subgraph, or the consequences of deleting one vertex.
TL;DR: Using a recent result of Chudnovsky, Seymour, and Sullivan, two bounds related to the Caccetta-Haggkvist Conjecture are slightly improved.
Abstract: Using a recent result of Chudnovsky, Seymour, and Sullivan, we slightly improve two bounds related to the Caccetta-Haggkvist Conjecture. Namely, we show that if $\alpha\geq 0.35312$, then each $n$-vertex digraph $D$ with minimum outdegree at least $\alpha n$ has a directed $3$-cycle. If $\beta\geq 0.34564$, then every $n$-vertex digraph $D$ in which the outdegree and the indegree of each vertex is at least $\beta n$ has a directed $3$-cycle.
TL;DR: It is shown that the generating function (in $n$) for the number of walks on the square lattice with steps $(1,1, (1,-1), (2,2)$ and $(2,-2)) satisfies a very special fifth order nonlinear recurrence relation in the region of the lattice.
Abstract: We show that the generating function (in $n$) for the number of walks on the square lattice with steps $(1,1), (1,-1), (2,2)$ and $(2,-2)$ from $(0,0)$ to $(2n,0)$ in the region $0 \leq y \leq w$ satisfies a very special fifth order nonlinear recurrence relation in $w$ that implies both its numerator and denominator satisfy a linear recurrence relation.
TL;DR: An LYM-type inequality for partial permutation is established, and Ku and Leader's conjecture on maximal $k$-uniform intersecting families of partial permutations is proved.
Abstract: An Erdős-Ko-Rado-type theorem was established by Bollobas and Leader for $q$-signed sets and by Ku and Leader for partial permutations. In this paper, we establish an LYM-type inequality for partial permutations, and prove Ku and Leader's conjecture on maximal $k$-uniform intersecting families of partial permutations. Similar results on general colored sets are presented.
TL;DR: Surprisingly, it shows that many Van der Waerden certificates, which must avoid repetitions in terms of arithmetic progressions, reveal strong regularities with respect to several other criteria.
Abstract: We present the Cyclic Zipper Method, a procedure to construct lower bounds for Van der Waerden numbers. Using this method we improved seven lower bounds. For natural numbers $r$, $k$ and $n$ a Van der Waerden certificate $W(r,k,n)$ is a partition of $\{1, \ldots, n\}$ into $r$ subsets, such that none of them contains an arithmetic progression of length $k$ (or larger). Van der Waerden showed that given $r$ and $k$, a smallest $n$ exists - the Van der Waerden number $W(r,k)$ - for which no certificate $W(r,k,n)$ exists. In this paper we investigate Van der Waerden certificates which have certain symmetrical and repetitive properties. Surprisingly, it shows that many Van der Waerden certificates, which must avoid repetitions in terms of arithmetic progressions, reveal strong regularities with respect to several other criteria. The Cyclic Zipper Method exploits these regularities. To illustrate these regularities, two techniques are introduced to visualize certificates.
TL;DR: In this article, it was shown that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs.
Abstract: Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc 4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.
TL;DR: This paper considers the problem $k-SBR, a version of SBR in which each symbol is allowed to appear up to $k$ times in each string, for some $k\geq 1$ and proposes a $\Theta(k)$-approximation algorithm running in time $O(n)$; compared to the previously known algorithm, this is an improvement by a factor of $\TheTA( k)$ in the approximation ratio.
Abstract: In the last decade there has been an ongoing interest in string comparison problems; to a large extend the interest was stimulated by genome rearrangement problems in computational biology but related problems appear in many other areas of computer science. Particular attention has been given to the problem of sorting by reversals (SBR): given two strings, $A$ and $B$, find the minimum number of reversals that transform the string $A$ into the string $B$ (a reversal $\rho(i,j)$, $i minimum common string partition problem (MCSP): given two strings, $A$ and $B$, find a minimum size partition of $A$ into substrings $P_1,\ldots,P_l$ (i.e., $A=P_1\ldots P_l$) and a partition of $B$ into substrings $Q_1,\ldots,Q_l$ such that $(Q_1,\ldots,Q_l)$ is a permutation of $(P_1,\ldots,P_l)$. Primarily the SBR problem has been studied for strings in which every symbol appears exactly once (that is, for permutations) and only recently attention has been given to the general case where duplicates of the symbols are allowed. In this paper we consider the problem $k$-SBR, a version of SBR in which each symbol is allowed to appear up to $k$ times in each string, for some $k\geq 1$. The main result of the paper is a $\Theta(k)$-approximation algorithm for $k$-SBR running in time $O(n)$; compared to the previously known algorithm for $k$-SBR, this is an improvement by a factor of $\Theta(k)$ in the approximation ratio, and by a factor of $\Theta(k)$ in the running time. We approach the $k$-SBR by finding an approximation for the $k$-MCSP first and then turning it into a solution for $k$-SBR. Crucial ingredients of our algorithm are the suffix tree data structure and a linear time algorithm for a special case of a disjoint set union problem.
TL;DR: Lower bounds for the distance $k$-irredundance number of graphs and trees are established and it is proved that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k-kn_1(T) $ for each tree $T=(V,E)$ with $n_1 (T)$ leaves.
Abstract: A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]
eq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$ . A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant . The $k$-irredundance number of $G$ , denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemanska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.
TL;DR: It is shown that there exists a natural construction of random directed animals on any directed graph together with a particles system – a gas model with nearest exclusion – that explains combinatorialy the formal link known between the density of the gas model and the generating function of directed animals counted according to the area.
Abstract: In this paper, we revisit the enumeration of directed animals using gas models. We show that there exists a natural construction of random directed animals on any directed graph together with a particles system – a gas model with nearest exclusion – that explains combinatorialy the formal link known between the density of the gas model and the generating function of directed animals counted according to the area. This provides some new methods to compute the generating function of directed animals counted according to area, and leads in the particular case of the square lattice to new combinatorial results and questions. A model of gas related to directed animals counted according to area and perimeter on any directed graph is also exhibited.
TL;DR: A combinatorial analysis is performed that gives an exact formula for the number of $(p,q,n)$-dipoles in the torus and double torus.
Abstract: There are many applications of the enumeration of maps in surfaces to other areas of mathematics and the physical sciences. In particular, in quantum field theory and string theory, there are many examples of occasions where it is necessary to sum over all the Feynman graphs of a certain type. In a recent paper of Constable et al. on pp-wave string interactions, they must sum over a class of Feynman graphs which are equivalent to what we call $(p,q,n)$-dipoles. In this paper we perform a combinatorial analysis that gives an exact formula for the number of $(p,q,n)$-dipoles in the torus (genus 1) and double torus (genus 2).
TL;DR: This note provides a quick, self-contained and completely elementary alternative proof of the same results of the coefficient-free cluster algebra of affine type A_1^{(1)}$.
Abstract: We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was given by P.Caldero and the author. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp. The original argument by P.Caldero and the author used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton. This note provides a quick, self-contained and completely elementary alternative proof of the same results.
TL;DR: In this article, it was shown that there is a jump in the possible speeds of a hereditary property of tournaments from polynomial to exponential speed, and the minimal exponential speed of the hereditary property was shown to be Ω(1+o(1))n.
Abstract: A collection of unlabelled tournaments ${\cal P}$ is called a hereditary property if it is closed under isomorphism and under taking induced sub-tournaments. The speed of ${\cal P}$ is the function $n \mapsto |{\cal P}_n|$, where ${\cal P}_n = \{T \in {\cal P} : |V(T)| = n\}$. In this paper, we prove that there is a jump in the possible speeds of a hereditary property of tournaments, from polynomial to exponential speed. Moreover, we determine the minimal exponential speed, $|{\cal P}_n| = c^{(1+o(1))n}$, where $c \simeq 1.47$ is the largest real root of the polynomial $x^3 = x^2 + 1$, and the unique hereditary property with this speed.
TL;DR: R rooted triangulations of a sphere with multiple holes are enumerated by the total number of edges and the length of each boundary component by W.T. Tutte.
Abstract: We enumerate rooted triangulations of a sphere with multiple holes by the total number of edges and the length of each boundary component. The proof relies on a combinatorial identity due to W.T. Tutte.
TL;DR: In this article, the authors used methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets.
Abstract: We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, $4\le n\le 7$. Combined with results of Esselmann this gives the classification of all compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, $n\ge 4$. Polytopes in dimensions $2$ and $3$ were classified by Poincare and Andreev.
TL;DR: The total domination number of G is shown to be at most $n/2, which generalizes a result by Lam and Wei stating that if G is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-2 vertices, then the total domination ofG is at most n/2.
Abstract: A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal L}$; it is a path. If $|P| \equiv 0 \, ( {\rm mod} \, 4)$ and either the two ends of $P$ are adjacent in $G$ to the same large vertex or the two ends of $P$ are adjacent to different, but adjacent, large vertices in $G$, we call $P$ a $0$-path. If $|P| \ge 5$ and $|P| \equiv 1 \, ( {\rm mod} \, 4)$ with the two ends of $P$ adjacent in $G$ to the same large vertex, we call $P$ a $1$-path. If $|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call $P$ a $3$-path. For $i \in \{0,1,3\}$, we denote the number of $i$-paths in $G$ by $p_i$. We show that the total domination number of $G$ is at most $(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if $G$ is a graph of order $n$ with minimum degree at least three, then the total domination of $G$ is at most $n/2$. It also generalizes a result by Lam and Wei stating that if $G$ is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-$2$ vertices, then the total domination of $G$ is at most $n/2$.
TL;DR: Using classical cycle index enumeration with recent results from analytic combinatorics, several statistical properties of random unlabeled outerplanar graphs on n vertices are investigated, for instance concerning connectedness, the chromatic number, and the number of edges.
Abstract: We determine the exact and asymptotic number of unlabeled outerplanar graphs . The exact number $g_{n}$ of unlabeled outerplanar graphs on $n$ vertices can be computed in polynomial time, and $g_{n}$ is asymptotically $g\, n^{-5/2}\rho^{-n}$, where $g\approx0.00909941$ and $\rho^{-1}\approx7.50360$ can be approximated. Using our enumerative results we investigate several statistical properties of random unlabeled outerplanar graphs on $n$ vertices, for instance concerning connectedness, the chromatic number, and the number of edges. To obtain the results we combine classical cycle index enumeration with recent results from analytic combinatorics.