Beyond the Shannon's Bound

Let G = (V,E) be a multigraph of maximum degree ∆. The edges of G can be colored with at most 3 2∆ colors by Shannon’s theorem. We study lower bounds on the size of subgraphs of G that can be colored with ∆ colors. Shannon’s Theorem gives a bound of ∆ ⌊ 3 2 ∆⌋ |E|. However, for ∆ = 3, Kamiński and Kowalik [7, 8] showed that there is a 3-edge-colorable subgraph of size at least 79 |E|, unless G has a connected component isomorphic to K3 + e (a K3 with an arbitrary edge doubled). Here we extend this line of research by showing that G has a ∆-edge colorable subgraph with at least ∆ ⌊ 3 2 ∆⌋−1 |E| edges, unless ∆ is even and G contains ∆ 2 K3 or ∆ is odd and G contains ∆−1 2 K3+e. Moreover, the subgraph and its coloring can be found in polynomial time. Our results have applications in approximation algorithms for the Maximum kEdge-Colorable 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, for every even k ≥ 4 we obtain a 2k+2 3k+2 -approximation and for every odd k ≥ 5 we get a 2k+1 3k -approximation. When 4 ≤ k ≤ 13 this improves over earlier algorithms due to Feige et al. [5].