Complete characterization of graphs with local total antimagic chromatic number 3

Abstract

A total labeling of a graph $G = (V, E)$ is said to be local total antimagic if it is a bijection $f: V\cup E \to\{1,\ldots ,|V|+|E|\}$ such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights where the induced weight of a vertex $v$, $w_f(v) = \sum f(e)$ with $e$ ranging over all the edges incident to $v$, and the induced weight of an edge $uv$ is $w_f(uv) = f(u) + f(v)$. The local total antimagic chromatic number of $G$, denoted by $\chi_{lt}(G)$, is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of $G$. In this paper, we first obtained general lower and upper bounds for $\chi_{lt}(G)$ and sufficient conditions to construct a graph $H$ with $k$ pendant edges and $\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}$. We then completely characterized graphs $G$ with $\chi_{lt}(G)=3$. Many families of (disconnected) graphs $H$ with $k$ pendant edges and $\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}$ are also obtained.