A note on local antimagic chromatic number of lexicographic product graphs

TL;DR

This paper investigates the local antimagic chromatic number of lexicographic product graphs, providing conditions for its bounds and exploring when it equals the product of the chromatic numbers of the component graphs.

Contribution

It establishes sufficient conditions for the local antimagic chromatic number of lexicographic product graphs to be bounded by the product of the individual numbers and explores cases where equality holds.

Findings

Derived bounds for $oldsymbol{oldsymbol{ ext{chi}}_{la}(G[H])}$

Provided examples where $oldsymbol{oldsymbol{ ext{chi}}_{la}(G[H]) = ext{chi}(G) ext{chi}(H)$

Formulated conjectures relating $oldsymbol{oldsymbol{ ext{chi}}_{la}(G[H])}$ to graph cycles

Abstract

Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in E(u)} f(e)$ and $E(u$) is the set of edges incident to $u$. A graph $G$ is called local antimagic if $G$ admits at least a local antimagic labeling. The local antimagic chromatic number, denoted $\chi_{la}(G)$, is the minimum number of induced colors taken over local antimagic labelings of $G$. Let $G$ and $H$ be two disjoint graphs. The graph $G[H]$ is obtained by the lexicographic product of $G$ and $H$. In this paper, we obtain sufficient conditions for $\chi_{la}(G[H])\leq \chi_{la}(G)\chi_{la}(H)$. Consequently, we give examples of $G$ and $H$ such that $\chi_{la}(G[H]) = \chi(G)\chi(H)$, where $\chi(G)$ is the chromatic number of $G$. We conjecture that (i) there are infinitely many graphs $G$ and $H$ such that $\chi_{la}(G[H])=\chi_{la}(G)\chi_{la}(H) = \chi(G)\chi(H)$, and (ii) for $k\ge 1$, $\chi_{la}(G[H]) = \chi(G)\chi(H)$ if and only if $\chi(G)\chi(H) = 2\chi(H) + \lceil\frac{\chi(H)}{k}\rceil$, where $2k+1$ is the length of a shortest odd cycle in $G$.