Equitable Neighbor Sum Distinguishing Edge Colorings of Some Graphs

Abstract

The edge coloring problem of graphs has interesting real life applications in the optimization and the network design, such as the file transfers in computer networks. A $k$-edge coloring of a graph $G$ is a mapping $\phi: E(G) \rightarrow \{1,\cdots,k\}$. Let $f(v)$ denote the sum of the colors on all the edges incident to $v$. A $k$-neighbor sum distinguishing edge coloring of $G$ is a $k$-edge coloring of $G$ such that for each edge $uv\in E(G)$, $f(u)\neq f(v)$. The neighbor sum distinguishing index of a graph $G$ is then the smallest $k$ for which $G$ admits a $k$-neighbor sum distinguishing edge coloring. In this paper, we study the equitable neighbor sum distinguishing edge coloring, it is a $k$-neighbor sum distinguishing edge coloring $\phi$ for which the number of edges in any two color classes of $\phi$ differ by at most one. The smallest value $k$ in such a coloring of $G$ is called equitable neighbor sum distinguishing index, denoted by $\overline{\chi^{e}_{\sum}}(G)$. Exact value of $\overline{\chi^{e}_{\sum}}(G)$ are determined for several classes of graphs, including cycles, fan graphs and theta graphs.