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 C 5 ) of order at least 3 , for all trees of order at least 3 .