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 $f:V\cup E\to \{1,\dots ,|V|+|E|\}$ 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.

arXiv comment18 pages, 8 figures. In this version, we corrected major mistakes in the characterization of graphs with ${\chi }_{lt}=3$