Maximum $Δ$-edge-colorable subgraphs of class II graphs

Details Maximum $Δ$-edge-colorable subgraphs of class II graphs Maximum $Δ$-edge-colorable subgraphs of class II graphs

2010-02-03

arxiv.org/abs/1002.0783v2

Computer Science

Discrete Mathematics

13 pages, 2 figures, the proof of the Lemma 1 is corrected

Scientific paper

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $\Delta$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $\Delta\geq3$ can be extended to a maximum $\Delta$-edge-colorable subgraph. Simple graphs have a maximum $\Delta$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $\Delta$-edge-colorable subgraph of a simple graph is always class I.

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