Cubic Graphs, Disjoint Matchings and Some Inequalities

For k = 2, 3 and a cubic graph G let νk(G) denote the size of a maximum k-edge-colorable subgraph of G. Mkrtchyan, Petrosyan and Vardanyan proved that ν2(G) ≥ 4 5 · |V (G)|, ν3(G) ≥ 7 6 · |V (G)| [13]. They were also able to show that ν2(G) + ν3(G) ≥ 2 · |V (G)| [3] and ν2(G) ≤ |V (G)|+2·ν3(G) 4 [13]. In the present work, we show that the last two inequalities imply the first two of them. Moreover, we show that ν2(G) ≥ α · |V (G)|+2·ν3(G) 4 , where α = 16 17 , if G is a cubic graph, α = 20 21 , if G is a cubic graph containing a perfect matching, α = 44 45 , if G is a bridgeless cubic graph. Finally, we investigate the parameters ν2(G) and ν3(G) in the class of claw-free cubic graphs. We improve the lower bounds for ν2(G) and ν3(G) for claw-free bridgeless cubic graphs to ν2(G) ≥ 29 30 · |V (G)|, ν3(G) ≥ 43 45 · |E(G)|. We also show that ν2(G) ≥ 35 36 · |V (G)| when n ≥ 48. On the basis of these inequalities we are able to improve the coefficient α for bridgeless claw-free cubic graphs.