On local antimagic chromatic number of lexicographic product graphs

Abstract

Let G = (V, E) be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection f : E → {1, 2, . . . , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have f + (u) = f + (v), where f + (u) = e∈E(u) f (e), and E(u) is the set of edges incident to u. Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex v is assigned the color f + (v). The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The lexicographic product of G and H, denoted G[H], is the graph with vertex set V (G) × V (H), and (u,In this paper, we obtained sharp upper bound of χ la (G[O n ]) where O n is a null graph of order n ≥ 1. Sufficient conditions for even regular bipartite and tripartite graphs G to have χ la (G) = 3 are also obtained. Consequently, we successfully determined the local antimagic chromatic number of infinitely many (connected and disconnected) regular graphs that partially support the existence of r-regular graph G of order p such that (i) χ la (G) = χ(G) = k, and (ii) χ la (G) = χ(G) + 1 = k for each possible r, p, k.

On local antimagic chromatic number of spider graphs

“…The lexicographic product G[H] of graphs G and H is a graph such that its vertex set is the cartesian product V (G) × V (H), and any two vertices (u, u ′ ) and (v, v ′ ) are adjcent in G[H] if and only if either uv ∈ E(G) or u = v and u ′ v ′ ∈ E(H). In [9], the first two authors studied the exact value of χ la (G[O n ]), where O n is a null graph of order n ≥ 2. Motivated by the above result, we investigate the sharp upper bound of χ la (G[H]) for any two disjoint non-null graphs G and H in this paper.…”