Vertex-distinguishing edge coloring of graphs

Paper

Abstract

Let

be an integer and let

be a nonempty simple graph. An \emph{edge-

-coloring}

is an assignment of colors from

to the edges of

such that no two adjacent edges receive the same color. For a vertex

, we write

for the set of colors assigned to the edges incident with

. The coloring

is called \emph{vertex-distinguishing} if

for every pair of distinct vertices

. A vertex-distinguishing edge-

-coloring exists if and only if

has at most one isolated vertex and no isolated edge. The least integer

for which such a coloring exists is called the \emph{vertex-distinguishing chromatic index} of

, denoted

. In 1997, Burris and Schelp conjectured that for every graph

with at most one isolated vertex and no isolated edge,

, where

is the natural lower bound required for a vertex-distinguishing coloring in

. In 2004, Balister, Kostochka, Li, and Schelp verified the conjecture for graphs

satisfying

and

. For graphs that do not satisfy these conditions, the best known general upper bound on

remains

, established in 1999 by Bazgan, Harkat-Benhamdine, Li, and Woźniak. In this paper, we prove that

, which represents a substantial improvement over the bound

whenever

. We further show that

, for all

-regular graphs

with