Every Graph Is Local Antimagic Total And Its Applications To Local Antimagic (Total) Chromatic Numbers

Abstract

A graph $G = (V, E)$ of order $p$ and size $q$ is said to be local antimagic if there exists a bijection $g:E(G) \to \{1,2,\ldots,q\}$ such that for any pair of adjacent vertices $u$ and $v$, $g^+(u)\ne g^+(v)$, where $g^+(u)=\sum_{uv\in E(G)} g(uv)$ is the induced vertex color of $u$ under $g$. We also say $G$ is local antimagic total if there exists a bijection $f: V\cup E \to\{1,2,\ldots ,p+q\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\not= w(v)$, where $w(u)= f(u) +\sum_{uv\in E(G)} f(uv)$ is the induced vertex weight of $u$ under $f$. The local antimagic (and local antimagic total) chromatic number of $G$, denoted $\chi_{la}(G)$ (and $\chi_{lat}(G)$), is the minimum number of distinct induced vertex colors (and weights) over all local antimagic (and local antimagic total) labelings of $G$. We also say a local antimagic total labeling is local super antimagic total if $f(v)\in\{1,2,\ldots,p\}$ for each $v\in V(G)$. In [Proof of a local antimagic conjecture, {\it Discrete Math. Theor. Comp. Sc.}, {\bf 20(1)} (2018), \#18], the author proved that every connected graph of order at least 3 is local antimagic. Using this result, we provide a very short proof that every graph is local antimagic total. We showed that there exists close relationship between $\chi_{la}(G \vee K_1)$ and $\chi_{lat}(G)$. A sufficient condition is also given for the corresponding local super antimagic total labeling. Sharp bounds of $\chi_{lat}(G)$ and close relationships between $\chi_{lat}(G)$ and $\chi_{la}(G \vee K_1)$ are found. Bounds of $\chi_{lat}(G-e)$ in terms of $\chi_{lat}(G)$ for a graph $G$ with an edge $e$ deleted are also obtained. These relationships are used to determine the exact values of $\chi_{lat}$ for many graphs $G$. We also conjecture that each graph $G$ of order at least 3 has $\chi_{lat}(G)\le \chi_{la}(G)$.