On twin edge mean colorings of graphs

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. In this paper, we introduce a new neighbor-distinguishing coloring called twin edge mean coloring. A proper edge coloring of G that uses colors from _k = {0,1,..., k − 1} is called a twin k-edge mean coloring of G if it induces a proper vertex coloring of G such that the color of each vertex υ of G is the average of the colors of the edges incident with υ, and is an integer. The minimum k for which G has a twin k-edge mean coloring is called the twin chromatic mean index of G and is denoted by $\chi _{tm}^\prime (G)$. First, we establish lower and upper bounds for $\chi _{tm}^\prime (G)$ under general or more specific assumptions. Then we determine the twin chromatic mean indices of paths, cycles, and stars.