On twin edge colorings in m-ary trees

Abstract

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. A twin k-edge coloring of G is a proper edge coloring of G that uses colors from ℤ_k and that induces a proper vertex coloring on G where the color of a vertex v is the sum (in ℤ_k) of the colors of the edges incident with v. The smallest integer k for which G has a twin k-edge coloring is the twin chromatic index of G and is denoted by χ′_t(G). In this paper, we study the twin edge colorings in m-ary trees for m ≥ 2; in particular, the twin chromatic indexes of full m-ary trees that are not stars, r-regular trees for even r ≥ 2, and generalized star graphs that are not paths nor stars are completely determined. Moreover, our results confirm the conjecture that χ′_t(G)≤Δ(G)+2 for every connected graph G (except C₅) of order at least 3, for all trees of order at least 3.