Local edge antimagic chromatic number of comb products involving path graph

Abstract

Let G = (V, E) be a graph with n vertices and no isolated vertices. A local edge antimagic labeling of G is a bijection f : V(G)→{1, 2, …, n} such that the weights of any two adjacent edges in G are distinct, where the weight of an edge in G is defined as the sum of the labels of its end vertices. Such a labeling induces a proper edge coloring of G, with edge weights serving as the colors. The local edge antimagic chromatic number of G, denoted χ′_lea(G), is the minimum number of colors used across all such labelings. In this paper, we investigate the local edge antimagic chromatic number of comb product graphs, focusing on the case where a path graph is combined with copies of other graphs—specifically paths, cycles, and ladders. The comb product of G and H, with respect to an assigned vertex, is constructed by taking one copy of G and |V(G)| copies of H and identifying the assigned vertex from the i-th copy of H to the i-th vertex of G.