4,367 research outputs found

    Connection Matrices and the Definability of Graph Parameters

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    In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers

    Conflict-Free Coloring of Planar Graphs

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    A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

    Normal edge-colorings of cubic graphs

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    A normal kk-edge-coloring of a cubic graph is an edge-coloring with kk colors having the additional property that when looking at the set of colors assigned to any edge ee and the four edges adjacent it, we have either exactly five distinct colors or exactly three distinct colors. We denote by χN(G)\chi'_{N}(G) the smallest kk, for which GG admits a normal kk-edge-coloring. Normal kk-edge-colorings were introduced by Jaeger in order to study his well-known Petersen Coloring Conjecture. More precisely, it is known that proving χN(G)5\chi'_{N}(G)\leq 5 for every bridgeless cubic graph is equivalent to proving Petersen Coloring Conjecture and then, among others, Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the larger class of all simple cubic graphs (not necessarily bridgeless), some interesting questions naturally arise. For instance, there exist simple cubic graphs, not bridgeless, with χN(G)=7\chi'_{N}(G)=7. On the other hand, the known best general upper bound for χN(G)\chi'_{N}(G) was 99. Here, we improve it by proving that χN(G)7\chi'_{N}(G)\leq7 for any simple cubic graph GG, which is best possible. We obtain this result by proving the existence of specific no-where zero Z22\mathbb{Z}_2^2-flows in 44-edge-connected graphs.Comment: 17 pages, 6 figure

    Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

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    We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
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