Local irregularity chromatic number of vertex shackle product of graphs

Abstract

Local irregular vertex labeling is one of graph labeling type that can be used as a tool for graph coloring. A mapping l is called local irregular vertex labeling if there are: (i) a mapping as vertex irregular k -labeling and a weight , for every where ; and (ii) apt(l) = min{max{li }. Thus, the labeling l induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local irregular chromatic number of G, denoted by is the minimum cardinality of the largest label over all such local irregular vertex labeling. In this paper, we determine the local irregular chromatic number of a vertex shackle product of graphs. The vertex shackle products of graphs, denoted by Shack (G, v, k), is the graph constructed from k copies of connected graph G and v as the linkage vertex.