Further results on local inclusive distance vertex irregularity strength of graphs

Fawwaz Fakhrurrozi Hadiputra (Master Program of Mathematics, Institut Teknologi Bandung, Bandung - Indonesia)
Eunike Setiawan (Master Program in Pure Mathematics, Imperial College London - United Kingdom)
Tita Khalis Maryati (Department of Mathematics Education, Universitas Islam Negeri Syarif Hidayatullah Jakarta, Ciputat - Indonesia)
Denny Riama Silaban (Department of Mathematics, Universitas Indonesia, Depok - Indonesia Center for Research Collaboration on Graph Theory and Combinatorics - Indonesia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 11, No 1 (2023): Electronic Journal of Graph Theory and Applications

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.