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On local antimagic chromatic number of the join of two special families of graphs -- II

Gee-Choon Lau, Wai Chee Shiu

Abstract

It is known that null graphs and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number is 3. Consequently, we also obtained many families of (possibly disconnected or regular) bipartite and tripartite graph with local antimagic chromatic number 3.

On local antimagic chromatic number of the join of two special families of graphs -- II

Abstract

It is known that null graphs and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number is 3. Consequently, we also obtained many families of (possibly disconnected or regular) bipartite and tripartite graph with local antimagic chromatic number 3.

Paper Structure

This paper contains 4 sections, 21 theorems, 18 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Matrix of size $(4n+1)\times 2k$
  3. Matrix of size $(4n+3)\times 2k$
  4. Conclusion and Open Problems

Key Result

Lemma 1.1

Let $G$ be a graph of size $q$. Suppose there is a local antimagic labeling of $G$ inducing a $2$-coloring of $G$ with colors $x$ and $y$, where $x<y$. Let $X$ and $Y$ be the sets of vertices colored $x$ and $y$, respectively, then $G$ is a bipartite graph with bipartition $(X,Y)$ and $|X|>|Y|$. Mor

Figures (7)

  • Figure 1: Graph $2P_2\vee O_2$.
  • Figure 2: Graph $8(P_2\vee O_4)$.
  • Figure 3: Graph $2(2P_2\vee O_{4})$.
  • Figure 4: Graph $G_{3,2}(4)$.
  • Figure 5: Graph $G_4(3,2)$ with a local antimagic 3-coloring.
  • ...and 2 more figures

Theorems & Definitions (39)

  • Lemma 1.1: LSN-IJMSI
  • Theorem 2.1
  • proof
  • Example 2.1
  • Theorem 2.2
  • proof
  • Example 2.2
  • Theorem 2.3
  • proof
  • Example 2.3
  • ...and 29 more