Beyond the Vizing's bound for at most seven colors

Let $G=(V,E)$ be a simple graph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\Delta +1$ colors by Vizing's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$ colors. Vizing's Theorem gives a bound of $\frac{\Delta}{\Delta+1}|E|$. This is known to be tight for cliques $K_{\Delta+1}$ when $\Delta$ is even. However, for $\Delta=3$ it was improved to $26/31|E|$ by Albertson and Haas [Parsimonious edge colorings, Disc. Math. 148, 1996] and later to $6/7|E|$ by Rizzi [Approximating the maximum 3-edge-colorable subgraph problem, Disc. Math. 309, 2009]. It is tight for $B_3$, the graph isomorphic to a $K_4$ with one edge subdivided. We improve previously known bounds for $\Delta\in{3,...,7}$, under the assumption that for $\Delta=3,4,6$ graph $G$ is not isomorphic to $B_3$, $K_5$ and $K_7$, respectively. For $\Delta \geq 4$ these are the first results which improve over the Vizing's bound. We also show a new bound for subcubic multigraphs not isomorphic to $K_3$ with one edge doubled. In the second part, we give 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, 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 7/9-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.