Decomposition of class II graphs into two class I graphs

TL;DR

This paper extends the decomposition of class II graphs into class I graphs to multigraphs and proves a variation of a conjecture, showing graphs with chromatic index (G)+k can be split into two class I subgraphs with specified maximum degrees.

Contribution

The authors generalize previous results to multigraphs and establish a new decomposition theorem for graphs with chromatic index (G)+k into two class I subgraphs.

Findings

Multigraphs with multiplicity (G) can be decomposed into a maximum (G)-edge-colorable subgraph and a subgraph with maximum degree at most (G)

Graphs with chromatic index (G)+k can be decomposed into two class I subgraphs with maximum degrees (G) and k

Extension of graph decomposition results to multigraphs and a variation of the conjecture for (G)+k graphs.

Abstract

Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $\Delta$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $\Delta(G)+k$ ($k\geq 1$) can be decomposed into a maximum $\Delta(G)$-edge-colorable subgraph (not necessarily class I) and a $k$-edge-colorable subgraph. In this paper, we first generalize their result to multigraphs and show that every multigraph $G$ with multiplicity $\mu$ can be decomposed into a maximum $\Delta(G)$-edge-colorable subgraph and a subgraph with maximum degree at most $\mu$. Then we prove that every graph $G$ with chromatic index $\Delta(G)+k$ can be decomposed into two class I subgraphs $H_1$ and $H_2$ such that $\Delta(H_1) = \Delta(G)$ and $\Delta(H_2) = k$, which is a variation of their conjecture.