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 $g:E(G)\to \{1,2,\dots ,q\}$ 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 $f:V\cup E\to \{1,2,\dots ,p+q\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\ne w(v)$, 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 $f(v)\in \{1,2,\dots ,p\}$ 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 ${\chi }_{lat}(G)\le {\chi }_{la}(G)$.

It is extended to 27 pages with many more new results

References in corpus (3)

• Proof of a local antimagic conjecture
• On a combination of the 1-2-3 Conjecture and the Antimagic Labelling Conjecture
• Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number

Cited by in corpus (1)

• On Integer Sequences Associated To Two Distinct Sums