Vertex Induced k−Edge Coloring and Vertex Incident k−Edge Coloring of Graphs

Abstract

Let k ≥ 2 be a natural number. Then the vertex induced k−edge
coloring number ψ′_vik(G) of a simple connected graph G = (V, E) is the highest number of colors needed to color the edges of a graph G such that the edges of the subgraph induced by the closed neighborhood N[v] of the vertex v ∈ V (G) receives
not more than k colors.
The vertex incident k−edge coloring number ψ′_vink(G) of a simple connected graph G = (V, E) is the highest number of colors required to color the edges of a graph G such that the edges incident to a vertex v in graph G receives not more than k colors. In this paper, we initiate the study on ψ′_vik(G) and ψ′_vink(G). We
also determine the exact values of ψ′_vik(G) and ψ′_vink(G) for k = 2 for some special graphs.

Key words and phrases. vertex induced k−edge coloring number, vertex incident k−edge coloring number, vertex induced 2−edge coloring number, vertex incident 2−edge coloring number.

2010 Mathematics Subject Classification. 05C15