On local antimagic chromatic number of lexicographic product graphs

TL;DR

This paper investigates the local antimagic chromatic number of lexicographic product graphs, providing bounds and conditions for specific classes of graphs, and explores the relationship between this number and the chromatic number.

Contribution

It establishes sharp upper bounds for the local antimagic chromatic number of lexicographic products with null graphs and characterizes conditions for certain regular bipartite and tripartite graphs.

Findings

Sharp upper bounds for $oldsymbol{ ext{chi}_{la}(G[O_n])}$ are derived.

Conditions under which regular bipartite and tripartite graphs have $oldsymbol{ ext{chi}_{la}(G)=3}$ are identified.

Infinite families of regular graphs with specific relationships between $oldsymbol{ ext{chi}_{la}(G)}$ and $oldsymbol{ ext{chi}(G)}$ are characterized.

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 $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$, 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 $\chi_{la}(G)$, is the minimum number of induced colors taken over local antimagic labeling of $G$. Let $G$ and $H$ be two vertex disjoint graphs. The {\it lexicographic product} of $G$ and $H$, denoted $G[H]$, is the graph with vertex set $V(G) \times V(H)$, and $(u,u')$ is adjacent to $(v,v')$ in $G[H]$ if $(u,v)\in E(G)$ or if $u=v$ and $u'v'\in E(H)$. In this paper, we obtained sharp upper bound of $\chi_{la}(G[O_n])$ where $O_n$ is a null graph of order $n\ge 1$. Sufficient conditions for even regular bipartite and tripartite graphs $G$ to have $\chi_{la}(G)=3$ 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) $\chi_{la}(G)=\chi(G)=k$, and (ii) $\chi_{la}(G)=\chi(G)+1=k$ for each possible $r,p,k$.