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arXiv · 2211.05930

Decomposition of class II graphs into two class I graphs

Yan Cao·Guangming Jing·Rong Luo·Vahan Mkrtchyan·Cun-Quan Zhang·Yue Zhao
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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 $Δ$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $Δ(G)+k$ ($k\geq 1$) can be decomposed into a maximum $Δ(G)$-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 $μ$ can be decomposed into a maximum $Δ(G)$-edge-colorable subgraph and a subgraph with maximum degree at most $μ$. Then we prove that every graph $G$ with chromatic index $Δ(G)+k$ can be decomposed into two class I subgraphs $H_1$ and $H_2$ such that $Δ(H_1) = Δ(G)$ and $Δ(H_2) = k$, which is a variation of their conjecture.

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Yan Cao, Guangming Jing, Rong Luo, Vahan Mkrtchyan, Cun-Quan Zhang, Yue Zhao. 2022-11-11. Decomposition of class II graphs into two class I graphs. https://arxiv.org/abs/2211.05930

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