Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach

Abstract

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if there is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we give three ways to construct a $(3m+2)\times (2k+1)$ matrix that meets certain properties for $m=1,3$ and $k\ge 1$. Consequently, we obtained many (disconnected) graphs of size $(3m+2)(2k+1)$ with local antimagic chromatic number 3.

Figures (7)

• Figure 1: The graph $9FB(1)$ for the graphs $FB(9)$ and $3FB(3)$.
• Figure 2: Graph $DF(1, 6)=DF(6)+FB(3)$.
• Figure 3: Graph $Pt(4)$.
• Figure 4: Graph $Pt(10)$.
• Figure 5: Graph $TB(10)$.

Theorems & Definitions (30)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Example 2.1
• Theorem 2.3
• proof
• Example 2.2
• Theorem 2.4
• proof