Approximating the maximum 3-edge-colorable subgraph problem

We offer the following structural result: every triangle-free graph G of maximum degree 3 has 3 matchings which collectively cover at least ( 1 − 2 3 γ o ( G ) ) of its edges, where γ o ( G ) denotes the odd girth of G . In particular, every triangle-free graph G of maximum degree 3 has 3 matchings which cover at least 13 / 15 of its edges. The Petersen graph, where we can 3-edge-color at most 13 of its 15 edges, shows this to be tight. We can also cover at least 6 / 7 of the edges of any simple graph of maximum degree 3 by means of 3 matchings; again a tight bound. fixed value of a parameter k ≥ 1 , the Maximum k -Edge-Colorable Subgraph Problem asks to k -edge-color the most of the edges of a simple graph received in input. The problem is known to be A P X -hard for all k ≥ 2 . However, approximation algorithms with approximation ratios tending to 1 as k goes to infinity are also known. At present, the best known performance ratios for the cases k = 2 and k = 3 were 5 / 6 and 4 / 5 , respectively. Since the proofs of our structural result are algorithmic, we obtain an improved approximation algorithm for the case k = 3 , achieving approximation ratio of 6 / 7 . Better bounds, and allowing also for parallel edges, are obtained for graphs of higher odd girth (e.g., a bound of 13 / 15 when the input multigraph is restricted to be triangle-free, and of 19 / 21 when C 5 ’s are also banned).