Equitable Edge Coloring of Some Join Graphs

Abstract

The notion of equitable coloring was introduced by Meyer in $1973$. Let $G\left(V,E\right)$ be a graph. For $k-$proper edge coloring $f$ of graph $G$, if $\left|\left|E_i\right|-\left|E_j\right|\right|\leq 1,$ $i, j=0,1,2,\cdots k-1$, where $E_i\left(G\right)$ is the set of edges of color $i$ in $G$, then $f$ is called a $k-$equitable edge coloring of graph $G$, and $\chi_e^\prime\left(G\right)=\min \{k|$ there is a k equitable edge-coloring of graph G$\}$ is called the equitable edge chromatic number of $G$. In this paper, we obtain the equitable edge chromatic number of the join graph of $P_l\vee K_{m,n}$ and $P_m\vee K_{1,n,n}$.