On local antimagic chromatic number of lexicographic product graphs

• Published in: Acta Mathematica Hungarica
• Main Author: Lau G.-C.; Shiu W.C.
• Format: Article
• Language: English
• Published: Springer Science and Business Media B.V. 2023
• Online Access: https://www.scopus.com/inward/record.uri?eid=2-s2.0-85149531589&doi=10.1007%2fs10474-023-01305-x&partnerID=40&md5=6ea3daa69d7d910baecb086e8da43164

Consider a simple connected graph G= (V, E) of order p and size q. For a bijection f: E→ { 1 , 2 , … , q} , let f+(u) = ∑ e∈E(u)f(e) where E(u) is the set of edges incident to u. We say f is a local antimagic labeling of G if for any two adjacent vertices u and v, we have f+(u) ≠ f+(v). The minimum number of distinct values of f+ taken over all local antimagic labeling of G is denoted by χla(G). Let G[H] be the lexicographic product of graphs G and H. In this paper, we obtain sharp upper bound for χla(G[On]) where On is a null graph of order n≥ 3. Sufficient conditions for even regular bipartite and tripartite graphs G to have χla(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 an r-regular graph G of order p such that (i) χla(G) = χ(G) = k, and (ii) χla(G) = χ(G) + 1 = k for each possible r, p, k. © 2023, Akadémiai Kiadó, Budapest, Hungary.