Showing papers in "Electronic Journal of Combinatorics in 2009"
Journal Article•
TL;DR: In this survey I have collected everything I could find on graph labelings techniques that have appeared in journals that are not widely available.
Abstract: A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.
2,367 citations
TL;DR: Some basic notions from combinatorial topology are applied to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings.
Abstract: We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short p ...
141 citations
TL;DR: The conjecture that equality holds for all values of $r,s and $t$ is proved and the conjecture is a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein.
Abstract: Given a collection of matchings ${\cal M} = (M_1, M_2, \ldots, M_q)$ (repetitions allowed), a matching $M$ contained in $\bigcup {\cal M}$ is said to be $s$- rainbow for ${\cal M}$ if it contains representatives from $s$ matchings $M_i$ (where each edge is allowed to represent just one $M_i$). Formally, this means that there is a function $\phi: M \to [q]$ such that $e \in M_{\phi(e)}$ for all $e \in M$, and $|Im(\phi)|\ge s$. Let $f(r,s,t)$ be the maximal $k$ for which there exists a set of $k$ matchings of size $t$ in some $r$-partite hypergraph, such that there is no $s$-rainbow matching of size $t$. We prove that $f(r,s,t)\ge 2^{r-1}(s-1)$, make the conjecture that equality holds for all values of $r,s$ and $t$ and prove the conjecture when $r=2$ or $s=t=2$. In the case $r=3$, a stronger conjecture is that in a $3$-partite $3$-graph if all vertex degrees in one side (say $V_1$) are strictly larger than all vertex degrees in the other two sides, then there exists a matching of $V_1$. This conjecture is at the same time also a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein. We prove a weaker version, in which the degrees in $V_1$ are at least twice as large as the degrees in the other sides. We also formulate a related conjecture on edge colorings of $3$-partite $3$-graphs and prove a similarly weakened version.
105 citations
TL;DR: The tropical Grassmannian parameterizing tropicalizations of linear spaces, while the Dressian parameterizes all planes in $\TP^{n-1}$ are studied and computed explicitly for n-1.
Abstract: The tropical Grassmannian parameterizes tropicalizations of ordinary linear spaces, while the Dressian parameterizes all tropical linear spaces in ${\Bbb T}{\Bbb P}^{n-1}$. We study these parameter spaces and we compute them explicitly for $n \leq 7$. Planes are identified with matroid subdivisions and with arrangements of trees. These representations are then used to draw pictures.
93 citations
TL;DR: It is proved that the on-line choice number of a graph $G$ on $n$ vertices is at most $\chi(G) \ln n+1$, and theon-line $b$-choice number of £G$ is at least ${e(G)-1\over e-1} (b-1+\ln n)+b$.
Abstract: This paper studies on-line list colouring of graphs. It is proved that the on-line choice number of a graph $G$ on $n$ vertices is at most $\chi(G) \ln n+1$, and the on-line $b$-choice number of $G$ is at most ${e\chi(G)-1\over e-1} (b-1+ \ln n)+b$. Suppose $G$ is a graph with a given $\chi(G)$-colouring of $G$. Then for any $(\chi(G) \ln n +1)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $L$-colouring of $G$. For any $({e\chi(G)-1\over e-1} (b-1+ \ln n)+b)$-assignment $L$ of $G$, we give a polynomial time algorithm which constructs an $(L,b)$-colouring of $G$. We then characterize all on-line $2$-choosable graphs. It is also proved that a complete bipartite graph of the form $K_{3,q}$ is on-line $3$-choosable if and only if it is $3$-choosable, but there are graphs of the form $K_{6,q}$ which are $3$-choosable but not on-line $3$-choosable. Some open questions concerning on-line list colouring are posed in the last section.
90 citations
TL;DR: The unitary Cayley graph associated to an arbitrary finite ring is studied, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, andvertex and edge chromatic number.
Abstract: We study the unitary Cayley graph associated to an arbitrary finite ring, determining precisely its diameter, girth, eigenvalues, vertex and edge connectivity, and vertex and edge chromatic number. We also compute its automorphism group, settling a question of Klotz and Sander. In addition, we classify all planar graphs and perfect graphs within this class.
84 citations
TL;DR: This work introduces a coloring game on graphs, in which each vertex of a graph $G$ owns a stack of $\ell_v{-}1$ erasers, and demonstrates that the two notions are not equivalent; $\ell-paintability is stronger than $\ell$-list colorability .
Abstract: We introduce a coloring game on graphs, in which each vertex $v$ of a graph $G$ owns a stack of $\ell_v{-}1$ erasers. In each round of this game the first player Mr. Paint takes an unused color, and colors some of the uncolored vertices. He might color adjacent vertices with this color – something which is considered "incorrect". However, Mrs. Correct is positioned next to him, and corrects his incorrect coloring, i.e., she uses up some of the erasers – while stocks (stacks) last – to partially undo his assignment of the new color. If she has a winning strategy, i.e., she is able to enforce a correct and complete final graph coloring, then we say that $G$ is $\ell$-paintable . Our game provides an adequate game-theoretic approach to list coloring problems. The new concept is actually more general than the common setting with lists of available colors. It could have applications in time scheduling, when the available time slots are not known in advance. We give an example that shows that the two notions are not equivalent; $\ell$-paintability is stronger than $\ell$-list colorability . Nevertheless, many deep theorems about list colorability remain true in the context of paintability. We demonstrate this fact by proving strengthened versions of classical list coloring theorems. Among the obtained extensions are paintability versions of Thomassen's, Galvin's and Shannon's Theorems.
83 citations
TL;DR: The basic properties of promotion and evacuation are surveyed and some generalizations of these two operations are discussed, which concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the minimal chains of any poset.
Abstract: Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schutzenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.
81 citations
TL;DR: The energy of unitary Cayley graph is obtained which extends a result of R. Balakrishnan for power of a prime and when they are hyperenergetic is determined and it is proved that the ratio of E(X_{n}) over 2(n-1) grows exponentially with k.
Abstract: In this note we obtain the energy of unitary Cayley graph $X_{n}$ which extends a result of R. Balakrishnan for power of a prime and also determine when they are hyperenergetic. We also prove that ${E(X_{n})\over 2(n-1)}\geq{2^{k}\over 4k}$, where $k$ is the number of distinct prime divisors of $n$. Thus the ratio ${E(X_{n})\over 2(n-1)}$, measuring the degree of hyperenergeticity of $X_{n}$, grows exponentially with $k$.
75 citations
TL;DR: An elementary combinatorial proof of a formula to express the higher partial derivatives of composite functions in terms of those of factor functions is presented.
Abstract: We present an elementary combinatorial proof of a formula to express the higher partial derivatives of composite functions in terms of those of factor functions.
70 citations
TL;DR: In this article, lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a given degree were given for strongly stable and square-free strongly stable ideals.
Abstract: We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a xed degree. The rst concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more nely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two dieren t directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a xed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coecien t eld.
TL;DR: It is proved that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, and (ii) $ Sp_S (G) = {\BF i) for any orientation $sigma” if andonly if G is a forest, where ${\bf i=\sqrt{-1}$.
Abstract: An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.
TL;DR: In this article, it was shown that some familiar combinatorial identities can be interpreted in terms of abelian powers and an asymptotic estimate of the number of Abelian Squares is given.
Abstract: An abelian square is a nonempty string of length $2n$ where the last $n$ symbols form a permutation of the first $n$ symbols. Similarly, an abelian $r$'th power is a concatenation of $r$ blocks, each of length $n$, where each block is a permutation of the first $n$ symbols. In this note we point out that some familiar combinatorial identities can be interpreted in terms of abelian powers. We count the number of abelian squares and give an asymptotic estimate of this quantity.
TL;DR: All connected cubic integral Cayley graphs are determined and some infinite families of connected integral Cayleys are introduced.
Abstract: Let $G$ be a non-trivial group, $S\subseteq G\setminus \{1\}$ and $S=S^{-1}:=\{s^{-1} \;|\; s\in S\}$. The Cayley graph of $G$ denoted by $\Gamma(S:G)$ is a graph with vertex set $G$ and two vertices $a$ and $b$ are adjacent if $ab^{-1}\in S$. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs.
TL;DR: This work asymptotically determines the size of the largest family F of subsets of subset of $\{1,\dots,n\}$ not containing a given poset P if the Hasse diagram of $P$ is a tree.
Abstract: We asymptotically determine the size of the largest family $\cal F$ of subsets of $\{1,\dots,n\}$ not containing a given poset $P$ if the Hasse diagram of $P$ is a tree. This is a qualitative generalization of several known results including Sperner's theorem.
TL;DR: If G does not contain the complete bipartite graph K_{t,s} as a subgraph, then the Kővari-Turan-Sos upper bound on the average degree of $G$ (the Zarankiewicz problem) is matched.
Abstract: Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővari-Turan-Sos upper bound on the average degree of $G$ (the Zarankiewicz problem).
TL;DR: All groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a ∈ S such that a / ∈ h S \ {a}i .
Abstract: Let G be a group and S a non-empty subset of G. If ab / ∈ S for any a, b ∈ S, then S is called sum-free. We show that if S is maximal by inclusion and no proper subset generates h Si then |S| ≤ 2. We determine all groups with a maximal (by inclusion) sum-free set of size at most 2 and all of size 3 where there exists a ∈ S such that a / ∈ h S \ {a}i .
TL;DR: Some related families of orthogonal polynomials of a discrete variable are presented and some of their applications in the study of (distance-regular) graphs and (completely regular) codes are surveyed.
Abstract: We present some related families of orthogonal polynomials of a discrete variable and survey some of their applications in the study of (distance-regular) graphs and (completely regular) codes. One of the main peculiarities of such orthogonal systems is their non-standard normalization condition, requiring that the square norm of each polynomial must equal its value at a given point of the mesh. For instance, when they are defined from the spectrum of a graph, one of these families is the system of the predistance polynomials which, in the case of distance-regular graphs, turns out to be the sequence of distance polynomials. The applications range from (quasi-spectral) characterizations of distance-regular graphs, walk-regular graphs, local distance-regularity and completely regular codes, to some results on representation theory.
TL;DR: In this article, a new family of Schur type functions plays the role of a dual object in the Cauchy identity and the dual Littlewood-Richardson co-expansion rule for the dual Schur functions is given.
Abstract: The double Schur functions form a distinguished basis of the ring ( xjja) which is a multiparameter generalization of the ring of symmetric functions ( x). The canonical comultiplication on ( x) is extended to ( xjja) in a natural way so that the double power sums symmetric functions are primitive elements. We calculate the dual Littlewood{Richardson coecien ts in two dieren t ways thus providing comultiplication rules for the double Schur functions. We also prove multiparameter analogues of the Cauchy identity. A new family of Schur type functions plays the role of a dual object in the identities. We describe some properties of these dual Schur functions including a combinatorial presentation and an expansion formula in terms of the ordinary Schur functions. The dual Littlewood{Richardson coecien ts provide a multiplication rule for the dual Schur functions.
TL;DR: In this article, a type B analogue of the derangement polynomials was given, based on the notion of excedances of type B introduced by Brenti.
Abstract: Based on the notion of excedances of type B introduced by Brenti, we give a type B analogue of the derangement polynomials. The connection between the derangement polynomials and Eulerian polynomials naturally extends to the type B case. Using this relation, we derive some basic properties of the derangement polynomials of type B, including the generating function formula, the Sturm sequence property, and the asymptotic normal distribution. We also show that the derangement polynomials are almost symmetric in the sense that the coefficients possess the spiral property.
TL;DR: It is shown that there exists a minimal percolating set of size $4n ^2/33 + o(n^2)$, but there does not exist one larger than $(n + 2)^2/6$.
Abstract: In standard bootstrap percolation, a subset $A$ of the grid $[n]^2$ is initially infected . A new site is then infected if at least two of its neighbours are infected, and an infected site stays infected forever. The set $A$ is said to percolate if eventually the entire grid is infected. A percolating set is said to be minimal if none of its subsets percolate. Answering a question of Bollobas, we show that there exists a minimal percolating set of size $4n^2/33 + o(n^2)$, but there does not exist one larger than $(n + 2)^2/6$.
TL;DR: This work considers a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete graph on Z_p, an odd prime, and its application to graph analysis.
Abstract: We consider a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete graph on $Z_p$, $p$ an odd prime.
TL;DR: In this paper, a q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph, which leads to a characterisation of random spaces over the Galois Fleld Fq that are invariant under the natural action of the inflnite group of invertible matrices with coe-cients from Fq.
Abstract: A q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces over the Galois fleld Fq that are invariant under the natural action of the inflnite group of invertible matrices with coe-cients from Fq.
TL;DR: If n is sufficiently large, then Maker can win the Hamiltonian cycle game within $n+1$ moves, and this bound is best possible and it settles a question of Hefetz, Krivelevich, Stojakovic and Szabo.
Abstract: We consider the fair Hamiltonian cycle Maker-Breaker game, played on the edge set of the complete graph $K_n$ on $n$ vertices. It is known that Maker wins this game if $n$ is sufficiently large. We are interested in the minimum number of moves needed for Maker in order to win the Hamiltonian cycle game, and in the smallest $n$ for which Maker has a winning strategy for this game. We prove the following results: (1) If $n$ is sufficiently large, then Maker can win the Hamiltonian cycle game within $n+1$ moves. This bound is best possible and it settles a question of Hefetz, Krivelevich, Stojakovic and Szabo; (2) If $n \geq 29$, then Maker can win the Hamiltonian cycle game. This improves the previously best bound of $600$ due to Papaioannou.
TL;DR: Some algebraic characterizations of k-walk-regularity are shown, which are based on the so-called local spectrum and predistance polynomials of G, where the number of cycles of length l rooted at a given vertex is a constant through all the graph.
Abstract: Considering a connected graph G with diameter D, we say that it is k-walkregular, for a given integer k (0 ≤ k ≤ D), if the number of walks of length l between any pair of vertices only depends on the distance between them, provided that this distance does not exceed k. Thus, for k = 0, this definition coincides with that of walk-regular graph, where the number of cycles of length l rooted at a given vertex is a constant through all the graph. In the other extreme, for k = D, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of k-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of G.
TL;DR: This paper proves that an exactly r-edge-colored complete graph of order $n$ has a rainbow matching of size $k(\ge 2)$ if $r \ge max\{{2k-3\choose 2}+2, {k-2\choosen 2}+(k- 2)(n-k+2)+2 \}$, $k \ge 2$, and $n \ge 1k+1$.
Abstract: An $r$-edge-coloring of a graph is an assignment of $r$ colors to the edges of the graph An exactly $r$-edge-coloring of a graph is an $r$-edge-coloring of the graph that uses all $r$ colors A matching of an edge-colored graph is called rainbow matching , if no two edges have the same color in the matching In this paper, we prove that an exactly $r$-edge-colored complete graph of order $n$ has a rainbow matching of size $k(\ge 2)$ if $r \ge max\{{2k-3\choose 2}+2, {k-2\choose 2}+(k-2)(n-k+2)+2 \}$, $k \ge 2$, and $n \ge 2k+1$ The bound on $r$ is best possible
TL;DR: This paper demonstrates that irrespectively of the choice of lists, quasirandom broadcasting is as fast as broadcasting in the randomized push model, up to lower order terms and reduces the number of random bits from O(log 2 n) to only ⌈log2 n⌉ per person.
Abstract: This paper addresses the following fundamental problem: Suppose that in a group of n people, where each person knows all other group members, a single person holds a piece of information that must be disseminated to everybody within the group. How should the people propagate the information so that after short time everyone is informed? The classical approach, known as the push model, requires that in each round, every informed person selects some other person in the group at random, whom it then informs. In a different model, known as the quasirandom push model, each person maintains a cyclic list, i.e., permutation, of all members in the group (for instance, a contact list of persons). Once a person is informed, it chooses a random member in its own list, and from that point onwards, it informs a new person per round, in the order dictated by the list. In this paper we show that with probability 1 − o(1) the quasirandom protocol informs everybody in (1 ± o(1))log 2 n+ln n rounds; furthermore we also show that this bound is tight. This result, together with previous work on the randomized push model, demonstrates that irrespectively of the choice of lists, quasirandom broadcasting is as fast as broadcasting in the randomized push model, up to lower order terms. At the same time it reduces the number of random bits from O(log 2 n) to only ⌈log2 n⌉ per person.
TL;DR: It is shown that given a positive integer $m$ there exist infinitely many pairs of non-isomorphic graphs with cospectral symmetry powers, based on theory of multidimensional extensions of coherent configurations.
Abstract: The symmetric $m$-th power of a graph is the graph whose vertices are $m$-subsets of vertices and in which two $m$-subsets are adjacent if and only if their symmetric difference is an edge of the original graph. It was conjectured that there exists a fixed $m$ such that any two graphs are isomorphic if and only if their $m$-th symmetric powers are cospectral. In this paper we show that given a positive integer $m$ there exist infinitely many pairs of non-isomorphic graphs with cospectral $m$-th symmetric powers. Our construction is based on theory of multidimensional extensions of coherent configurations.
TL;DR: The characteristic polynomials for all graphs on 12 vertices are found, and statistics related to the number of cospectral graphs are reported.
Abstract: We found the characteristic polynomials for all graphs on 12 vertices, and report statistics related to the number of cospectral graphs.
TL;DR: This work provides two short proofs of the following statement: nearly optimal bounds on the entrywise stability of the first eigenvector of (Erdős-Renyi) Random graphs.
Abstract: Let $G$ be a graph randomly selected from ${\bf G}_{n, p}$, the space of Erdős-Renyi Random graphs with parameters $n$ and $p$, where $p \geq {\log^6 n\over n}$. Also, let $A$ be the adjacency matrix of $G$, and $v_1$ be the first eigenvector of $A$. We provide two short proofs of the following statement: For all $i \in [n]$, for some constant $c>0$ $$\left|v_1(i) - {1\over\sqrt{n}}\right| \leq c {1\over\sqrt{n}} {\log n\over\log (np)} \sqrt{{\log n\over np}}$$ with probability $1 - o(1)$. This gives nearly optimal bounds on the entrywise stability of the first eigenvector of (Erdős-Renyi) Random graphs. This question about entrywise bounds was motivated by a problem in unsupervised spectral clustering. We make some progress towards solving that problem.