1,457 research outputs found

    b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

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    A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic

    Dynamic Chromatic Number of Bipartite Graphs

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    A dynamic coloring of a graph G is a proper vertex coloring such that for every vertex v Î V(G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a dynamic coloring with k colors, is called the dynamic chromatic number of G and denoted by c2(G). Montgomery conjectured that for every r-regular graph G, c2(G)-c(G) ≤ 2 . Finding an optimal upper bound for c2(G)-c(G) seems to be an intriguing problem. We show that there is a constant d such that every bipartite graph G with d(G) ³ d , has c2(G)-c(G) ≤ 2é(D(G))/(d(G))ù. It was shown that c2(G)-c(G) ≤ a' (G) +k* . Also, c2(G)-c(G) ≤ a(G) +k* . We prove that if G is a simple graph with d(G)>2, then c2(G)-c(G) ≤ (a' (G)+w(G) )/2 +k* . Among other results, we prove that for a given bipartite graph G=[X,Y], determining whether G has a dynamic 4-coloring l : V (G)®{a, b, c, d} such that a, b are used for part X and c, d are used for part Y is NP-complete

    b-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

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    no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic

    Dynamic Chromatic Number of Regular Graphs

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    A dynamic coloring of a graph GG is a proper coloring such that for every vertex vV(G)v\in V(G) of degree at least 2, the neighbors of vv receive at least 2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}. PhD thesis, West Virginia University, 2001.] that if GG is a kk-regular graph, then χ2(G)χ(G)2\chi_2(G)-\chi(G)\leq 2. In this paper, we prove that if GG is a kk-regular graph with χ(G)4\chi(G)\geq 4, then χ2(G)χ(G)+α(G2)\chi_2(G)\leq \chi(G)+\alpha(G^2). It confirms the conjecture for all regular graph GG with diameter at most 2 and χ(G)4\chi(G)\geq 4. In fact, it shows that χ2(G)χ(G)1\chi_2(G)-\chi(G)\leq 1 provided that GG has diameter at most 2 and χ(G)4\chi(G)\geq 4. Moreover, we show that for any kk-regular graph GG, χ2(G)χ(G)6lnk+2\chi_2(G)-\chi(G)\leq 6\ln k+2. Also, we show that for any nn there exists a regular graph GG whose chromatic number is nn and χ2(G)χ(G)1\chi_2(G)-\chi(G)\geq 1. This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A. Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In press].Comment: 8 page

    Bipartite induced density in triangle-free graphs

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    We prove that any triangle-free graph on nn vertices with minimum degree at least dd contains a bipartite induced subgraph of minimum degree at least d2/(2n)d^2/(2n). This is sharp up to a logarithmic factor in nn. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of n/dn/d and (2+o(1))n/logn(2+o(1))\sqrt{n/\log n} as nn\to\infty. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most O(min{n,(nlogn)/d})O(\min\{\sqrt{n},(n\log n)/d\}) as nn\to\infty. Relatedly, we also make two conjectures. First, any triangle-free graph on nn vertices has fractional chromatic number at most (2+o(1))n/logn(\sqrt{2}+o(1))\sqrt{n/\log n} as nn\to\infty. Second, any triangle-free graph on nn vertices has list chromatic number at most O(n/logn)O(\sqrt{n/\log n}) as nn\to\infty.Comment: 20 pages; in v2 added note of concurrent work and one reference; in v3 added more notes of ensuing work and a result towards one of the conjectures (for list colouring
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