On join product and local antimagic chromatic number of regular 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-s2.0-85147375303&doi=10.1007%2fs10474-023-01298-7&partnerID=40&md5=5161cc053d90a45b44bbb43def593731

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 χla(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 join graph of G and H, denoted G∨ H, is the graph with V(G∨ H) = V(G) ∪ V(H) and E(G∨ H) = E(G) ∪ E(H) ∪ { uv∣ u∈ V(G) , v∈ V(H) }. In this paper, we investigated χla(G∨ H). Consequently, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers. © 2023, Akadémiai Kiadó, Budapest, Hungary.