630 research outputs found

    Entropy and Graphs

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    The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J. K\"{o}rner in 1973. Although the notion of graph entropy has its roots in information theory, it was proved to be closely related to some classical and frequently studied graph theoretic concepts. For example, it provides an equivalent definition for a graph to be perfect and it can also be applied to obtain lower bounds in graph covering problems. In this thesis, we review and investigate three equivalent definitions of graph entropy and its basic properties. Minimum entropy colouring of a graph was proposed by N. Alon in 1996. We study minimum entropy colouring and its relation to graph entropy. We also discuss the relationship between the entropy and the fractional chromatic number of a graph which was already established in the literature. A graph GG is called \emph{symmetric with respect to a functional FG(P)F_G(P)} defined on the set of all the probability distributions on its vertex set if the distribution PP^* maximizing FG(P)F_G(P) is uniform on V(G)V(G). Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we prove that vertex transitive graphs are symmetric with respect to graph entropy. Furthermore, we show that a bipartite graph is symmetric with respect to graph entropy if and only if it has a perfect matching. As a generalization of this result, we characterize some classes of symmetric perfect graphs with respect to graph entropy. Finally, we prove that the line graph of every bridgeless cubic graph is symmetric with respect to graph entropy.Comment: 89 pages, 4 figures, MMath Thesi

    S-Packing Colorings of Cubic Graphs

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    Given a non-decreasing sequence S=(s_1,s_2,,s_k)S=(s\_1,s\_2, \ldots, s\_k) of positive integers, an {\em SS-packing coloring} of a graph GG is a mapping cc from V(G)V(G) to {s_1,s_2,,s_k}\{s\_1,s\_2, \ldots, s\_k\} such that any two vertices with color s_is\_i are at mutual distance greater than s_is\_i, 1ik1\le i\le k. This paper studies SS-packing colorings of (sub)cubic graphs. We prove that subcubic graphs are (1,2,2,2,2,2,2)(1,2,2,2,2,2,2)-packing colorable and (1,1,2,2,3)(1,1,2,2,3)-packing colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we provide an example of a cubic graph of order 3838 which is not (1,2,,12)(1,2,\ldots,12)-packing colorable

    On uniquely packable trees

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    An ii-packing in a graph GG is a set of vertices that are pairwise distance more than ii apart. A \emph{packing colouring} of GG is a partition X={X1,X2,,Xk}X=\{X_{1},X_{2},\ldots,X_{k}\} of V(G)V(G) such that each colour class XiX_{i} is an ii-packing. The minimum order kk of a packing colouring is called the packing chromatic number of GG, denoted by χρ(G)\chi_{\rho}(G). In this paper we investigate the existence of trees TT for which there is only one packing colouring using χρ(T)\chi_\rho(T) colours. For the case χρ(T)=3\chi_\rho(T)=3, we completely characterise all such trees. As a by-product we obtain sets of uniquely 33-χρ\chi_\rho-packable trees with monotone χρ\chi_{\rho}-coloring and non-monotone χρ\chi_{\rho}-coloring respectively

    A study on exponential-size neighborhoods for the bin packing problem with conflicts

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    We propose an iterated local search based on several classes of local and large neighborhoods for the bin packing problem with conflicts. This problem, which combines the characteristics of both bin packing and vertex coloring, arises in various application contexts such as logistics and transportation, timetabling, and resource allocation for cloud computing. We introduce O(1)O(1) evaluation procedures for classical local-search moves, polynomial variants of ejection chains and assignment neighborhoods, an adaptive set covering-based neighborhood, and finally a controlled use of 0-cost moves to further diversify the search. The overall method produces solutions of good quality on the classical benchmark instances and scales very well with an increase of problem size. Extensive computational experiments are conducted to measure the respective contribution of each proposed neighborhood. In particular, the 0-cost moves and the large neighborhood based on set covering contribute very significantly to the search. Several research perspectives are open in relation to possible hybridizations with other state-of-the-art mathematical programming heuristics for this problem.Comment: 26 pages, 8 figure

    Packing chromatic vertex-critical graphs

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    The packing chromatic number χρ(G)\chi_{\rho}(G) of a graph GG is the smallest integer kk such that the vertex set of GG can be partitioned into sets ViV_i, i[k]i\in [k], where vertices in ViV_i are pairwise at distance at least i+1i+1. Packing chromatic vertex-critical graphs, χρ\chi_{\rho}-critical for short, are introduced as the graphs GG for which χρ(Gx)<χρ(G)\chi_{\rho}(G-x) < \chi_{\rho}(G) holds for every vertex xx of GG. If χρ(G)=k\chi_{\rho}(G) = k, then GG is kk-χρ\chi_{\rho}-critical. It is shown that if GG is χρ\chi_{\rho}-critical, then the set {χρ(G)χρ(Gx): xV(G)}\{\chi_{\rho}(G) - \chi_{\rho}(G-x):\ x\in V(G)\} can be almost arbitrary. The 33-χρ\chi_{\rho}-critical graphs are characterized, and 44-χρ\chi_{\rho}-critical graphs are characterized in the case when they contain a cycle of length at least 55 which is not congruent to 00 modulo 44. It is shown that for every integer k2k\ge 2 there exists a kk-χρ\chi_{\rho}-critical tree and that a kk-χρ\chi_{\rho}-critical caterpillar exists if and only if k7k\le 7. Cartesian products are also considered and in particular it is proved that if GG and HH are vertex-transitive graphs and diam(G)+diam(H)χρ(G){\rm diam(G)} + {\rm diam}(H) \le \chi_{\rho}(G), then GHG\,\square\, H is χρ\chi_{\rho}-critical
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