# Institute of Mathematics

## On $g_c$-colorings of nearly bipartite graphs

#### Yuzhuo Zhang, Xia Zhang

###### Received September 11, 2016.   First published February 8, 2018.

Abstract:  Let $G$ be a simple graph, let $d(v)$ denote the degree of a vertex $v$ and let $g$ be a nonnegative integer function on $V(G)$ with $0\leq g(v)\leq d(v)$ for each vertex $v\in\nobreak V(G)$. A $g_c$-coloring of $G$ is an edge coloring such that for each vertex $v\in V(G)$ and each color $c$, there are at least $g(v)$ edges colored $c$ incident with $v$. The $g_c$-chromatic index of $G$, denoted by $\chi'_{g_c}(G)$, is the maximum number of colors such that a $g_c$-coloring of $G$ exists. Any simple graph $G$ has the $g_c$-chromatic index equal to $\delta_g(G)$ or $\delta_g(G)-1$, where $\delta_g(G)= \min_{v\in V(G)}\lfloor{d(v)}/{g(v)}\rfloor$. A graph $G$ is nearly bipartite, if $G$ is not bipartite, but there is a vertex $u\in V(G)$ such that $G-u$ is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph $G$ to have $\chi'_{g_c}(G)=\delta_g(G)$. Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.
Keywords:  edge coloring; nearly bipartite graph; edge covering coloring; $g_c$-coloring; edge cover decomposition
Classification MSC:  05C15
DOI:  10.21136/CMJ.2018.0477-16

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Affiliations:   Yuzhuo Zhang, Xia Zhang (corresponding author), School of Mathematics and Statistics, Shandong Normal University, 88 Wenhua E Rd, Lixia, Jinan 250358, P. R. China, e-mail: pandarhz@sina.com

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