Beyond the Vizing's bound for at most seven colors
نویسندگان
چکیده
Let G = (V,E) be a simple graph of maximum degree ∆. The edges of G can be colored with at most ∆ + 1 colors by Vizing’s theorem. We study lower bounds on the size of subgraphs of G that can be colored with ∆ colors. Vizing’s Theorem gives a bound of ∆ ∆+1 |E|. This is known to be tight for cliques K∆+1 when ∆ is even. However, for ∆ = 3 it was improved to 26 31 |E| by Albertson and Haas [Parsimonious edge colorings, Disc. Math. 148, 1996] and later to 67 |E| by Rizzi [Approximating the maximum 3-edge-colorable subgraph problem, Disc. Math. 309, 2009]. It is tight for B3, the graph isomorphic to a K4 with one edge subdivided. We improve previously known bounds for ∆ ∈ {3, . . . , 7}, under the assumption that for ∆ = 3, 4, 6 graph G is not isomorphic to B3, K5 and K7, respectively. For ∆ ≥ 4 these are the first results which improve over the Vizing’s bound. We also show a new bound for subcubic multigraphs not isomorphic to K3 with one edge doubled. In the second part, we give approximation algorithms for the Maximum k-EdgeColorable Subgraph problem, where given a graph G (without any bound on its maximum degree or other restrictions) one has to find a k-edge-colorable subgraph with maximum number of edges. In particular, when G is simple for k = 3, 4, 5, 6, 7 we obtain approximation ratios of 13 15 , 9 11 , 19 22 , 23 27 and 22 25 , respectively. We also present a 79 -approximation for k = 3 when G is a multigraph. The approximation algorithms follow from a new general framework that can be used for any value of k.
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ورودعنوان ژورنال:
- SIAM J. Discrete Math.
دوره 28 شماره
صفحات -
تاریخ انتشار 2014