On the Vertex-Distinguishing Edge Chromatic Number of 𝐏𝐦 ∨ 𝐂𝐧

Distinguishing Edge Chromatic Number

An adjacent vertex distinguishing edge-coloring or an avd-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree ∆ and with no isolated edges has an avd-coloring with at most ∆ + 300 colors, provided that ∆ > 1020. AMS Subject Classification: 05C15

Delta+300 is a bound on the adjacent vertex distinguishing edge chromatic number

The adjacent vertex-distinguishing total chromatic number of 1-tree

Let G = (V (G), E(G)) be a simple graph and T (G) be the set of vertices and edges of G. Let C be a k−color set. A (proper) total k−coloring f of G is a function f : T (G) −→ C such that no adjacent or incident elements of T (G) receive the same color. For any u ∈ V (G), denote C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}. The total k−coloring f of G is called the adjacent vertex-distinguishing if C(u) 6=...

The Distinguishing Chromatic Number

In this paper we define and study the distinguishing chromatic number, χD(G), of a graph G, building on the work of Albertson and Collins who studied the distinguishing number. We find χD(G) for various families of graphs and characterize those graphs with χD(G) = |V (G)|, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks’ Theorem for both the...

Bounds on the Distinguishing Chromatic Number

Collins and Trenk define the distinguishing chromatic number χD(G) of a graph G to be the minimum number of colors needed to properly color the vertices of G so that the only automorphism of G that preserves colors is the identity. They prove results about χD(G) based on the underlying graphG. In this paper we prove results that relate χD(G) to the automorphism group of G. We prove two upper bo...