1 research outputs found
Beyond the Vizing's bound for at most seven colors
Let be a simple graph of maximum degree . The edges of
can be colored with at most colors by Vizing's theorem. We study
lower bounds on the size of subgraphs of that can be colored with
colors.
Vizing's Theorem gives a bound of . This is known
to be tight for cliques when is even. However, for
it was improved to by Albertson and Haas [Parsimonious
edge colorings, Disc. Math. 148, 1996] and later to by Rizzi
[Approximating the maximum 3-edge-colorable subgraph problem, Disc. Math. 309,
2009]. It is tight for , the graph isomorphic to a with one edge
subdivided.
We improve previously known bounds for , under the
assumption that for graph is not isomorphic to ,
and , respectively. For these are the first results which
improve over the Vizing's bound. We also show a new bound for subcubic
multigraphs not isomorphic to 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.Comment: 34 page