Further results on local inclusive distance vertex irregularity strength of graphs

Abstract

Let G = ( V , E ) be a simple undirected graph. A labeling f : V ( G )→{1, …, k } is a local inclusive d -distance vertex irregular labeling of G if every adjacent vertices x , y ∈ V ( G ) have distinct weights, with the weight w ( x ), x ∈ V ( G ) is the sum of every labels of vertices whose distance from x is at most d . The local inclusive d -distance vertex irregularity strength of G , lidis ( G ) , is the least number k for which there exists a local inclusive d -distance vertex irregular labeling of G . In this paper, we prove a conjecture on the local inclusive d -distance vertex irregularity strength for d = 1 for tree and we generalize the result for block graph using the clique number. Furthermore, we present several results for multipartite graphs and we also observe the relationship with chromatic number.