Equitable Strong Edge Coloring of the Joins of Paths and Cycles

Abstract:

      For a proper edge coloring $c$ of a graph $G$, if the sets of colors of adjacent vertices are distinct, the edge coloring $c$ is called an adjacent strong edge coloring of $G$. Let $c_i$ be the number of edges colored by $i$. If $|c_i-c_j|\le 1$ for any two colors $i$ and $j$, then $c$ is an equitable edge coloring of $G$. The coloring $c$ is an equitable adjacent strong edge coloring of $G$ if it is both adjacent strong edge coloring and equitable edge coloring. The least number of colors of such a coloring $c$ is called the equitable adjacent strong chromatic index of $G$. In this paper, we determine the equitable adjacent strong chromatic index of the joins of paths and cycles. Precisely, we show that the equitable adjacent strong chromatic index of the joins of paths and cycles is equal to the maximum degree plus one or two.