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Vertex V (vertex + v)
Selected AbstractsFamilies of pairs of graphs with a large number of common cardsJOURNAL OF GRAPH THEORY, Issue 2 2010Andrew Bowler Abstract The vertex-deleted subgraph G,v, obtained from the graph G by deleting the vertex v and all edges incident to v, is called a card of G. The deck of G is the multiset of its unlabelled vertex-deleted subgraphs. The number of common cards of G and H (or between G and H) is the cardinality of the multiset intersection of the decks of G and H. In this article, we present infinite families of pairs of graphs of order n , 4 that have at least common cards; we conjecture that these, along with a small number of other families constructed from them, are the only pairs of graphs having this many common cards, for sufficiently large n. This leads us to propose a new stronger version of the Reconstruction Conjecture. In addition, we present an infinite family of pairs of graphs with the same degree sequence that have common cards, for appropriate values of n, from which we can construct pairs having slightly fewer common cards for all other values of n,10. We also present infinite families of pairs of forests and pairs of trees with and common cards, respectively. We then present new families that have the maximum number of common cards when one graph is connected and the other disconnected. Finally, we present a family with a large number of common cards, where one graph is a tree and the other unicyclic, and discuss how many cards are required to determine whether a graph is a tree. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 146,163, 2010 [source] Coloring quasi-line graphsJOURNAL OF GRAPH THEORY, Issue 1 2007Maria Chudnovsky Abstract A graph G is a quasi-line graph if for every vertex v, the set of neighbors of v can be expressed as the union of two cliques. The class of quasi-line graphs is a proper superset of the class of line graphs. A theorem of Shannon's implies that if G is a line graph, then it can be properly colored using no more than 3/2 ,(G) colors, where ,(G) is the size of the largest clique in G. In this article, we extend this result to all quasi-line graphs. We also show that this bound is tight. © 2006 Wiley Periodicals, Inc. J Graph Theory [source] Nowhere-zero 3-flows in locally connected graphsJOURNAL OF GRAPH THEORY, Issue 3 2003Hong-Jian Lai Abstract Let G be a graph. For each vertex v ,V(G), Nv denotes the subgraph induces by the vertices adjacent to v in G. The graph G is locally k -edge-connected if for each vertex v ,V(G), Nv is k -edge-connected. In this paper we study the existence of nowhere-zero 3-flows in locally k -edge-connected graphs. In particular, we show that every 2-edge-connected, locally 3-edge-connected graph admits a nowhere-zero 3-flow. This result is best possible in the sense that there exists an infinite family of 2-edge-connected, locally 2-edge-connected graphs each of which does not have a 3-NZF. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 211,219, 2003 [source] Some results about f -critical graphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Guizhen Liu Abstract An f -coloring of a multigraph G is a coloring of the edges of E such that each color appears at each vertex v , V at most f(v) times. The minimum number of colors needed to f -color G is called the f -chromatic index of G and is denoted by ,,f(G). Various scheduling problems on networks are reduced to finding an f -coloring of a multigraph. Any simple graph G has f -chromatic index equal to ,f(G) or ,f(G)+ 1, where ,f(G) = max v,V{, ,} and d(v) is the degree of vertex v. A connected graph G is called f -critical if ,,f(G)=,f(G)+1 and ,,f(G)=,f(G,e) < ,,f(G) for any edge e , E. Some results about f -critical graphs are given. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(3), 197,202 2007 [source] The fractional matching numbers of graphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2002Yan Liu Abstract A fractional matching of a graph G is a function f that assigns to each edge a number in [0, 1] such that, for each vertex v, , f(e) , 1, where the sum is taken over all edges incident to v. The fractional matching number of G is the supremum of ,e,E(G)f(e) over all fractional matchings f. In this paper, we provide a new formula for calculating the fractional matching numbers of graphs using the Gallai,Edmonds Structure Theorem. Thus, we characterize graphs for which the fractional matching number equals the matching number and graphs for which the fractional matching number is the maximum possible (one-half the number of vertices). © 2002 Wiley Periodicals, Inc. [source] The ultracenter and central fringe of a graphNETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2001Gary Chartrand Abstract The central distance of a central vertex v in a connected graph G with rad G < diam G is the largest nonnegative integer n such that whenever x is a vertex with d(v, x) , n then x is also a central vertex. The subgraph induced by those central vertices of maximum central distance is the ultracenter of G. The subgraph induced by the central vertices having central distance 0 is the central fringe of G. For a given graph G, the smallest order of a connected graph H is determined whose ultracenter is isomorphic to G but whose center is not G. For a given graph F, we determine the smallest order of a connected graph H whose central fringe is isomorphic to G but whose center is not G. © 2001 John Wiley & Sons, Inc. [source] Rapid mixing of Gibbs sampling on graphs that are sparse on averageRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2009Elchanan Mossel Abstract Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd,s-Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 - o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature ,, the mixing time of Gibbs sampling is at least n1+,(1/log log n). Recall that the Ising model with inverse temperature , defined on a graph G = (V,E) is the distribution over {±}Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of , or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < , and the Ising model defined on G (n, d/n), there exists a ,d > 0, such that for all , < ,d with probability going to 1 as n ,,, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd,s-Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub-graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and , < ,d, where d tanh(,d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source] | ||||||||||