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Beyond the Shannon's Bound

Let $G=(V,E)$ be a multigraph of maximum degree $Δ$. The edges of $G$ can be colored with at most $\frac{3}{2}Δ$ colors by Shannon's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $Δ$ colors. Shannon's Theorem gives a bound of $\fracΔ{\lfloor\frac{3}{2}Δ\rfloor}|E|$. However, for $Δ=3$, Kamiński and Kowalik [SWAT'10] showed that there is a 3-edge-colorable subgraph of size at least $\frac{7}{9}|E|$, unless $G$ has a connected component isomorphic to $K_3+e$ (a $K_3$ with an arbitrary edge doubled). Here we extend this line of research by showing that $G$ has a $Δ$-edge colorable subgraph with at least $\fracΔ{\lfloor\frac{3}{2}Δ\rfloor-1}|E|$ edges, unless $Δ$ is even and $G$ contains $\fracΔ{2}K_3$ or $Δ$ is odd and $G$ contains $\frac{Δ-1}{2}K_3+e$. Moreover, the subgraph and its coloring can be found in polynomial time. Our results have applications in approximation algorithms for the Maximum $k$-Edge-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 $k \ge 4$ we obtain a $\frac{2k+2}{3k+2}$-approximation and for every odd $k\ge 5$ we get a $\frac{2k+1}{3k}$-approximation. When $4\le k \le 13$ this improves over earlier algorithms due to Feige et al. [APPROX'02]