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The half-monochromatic colorings of plane graphs with even polygonal faces

Kazuhiro Ichihara, Yuha Tamura

TL;DR

The paper addresses finding the maximum number of colors in half-monochromatic colorings of plane graphs with even polygonal faces, extending the known result for plane quadrangulations. It develops a method based on dividing systems in the medial graph $M(G)$ and the division tree $T_\Lambda$ to connect color capacity to the independence number $\alpha(G)$ and proves the bound $\chi_f(G) \le \frac{3}{2}\alpha(G)$. A corollary shows $\chi_f(G)=\lambda$, where $\lambda$ is the maximum number of regions determined by a dividing system, and the construction attains equality in cases achieving the bound. The approach also confirms that a half-monochromatic coloring exists for all such graphs and reinforces the link between planar coloring constraints and classical graph invariants.

Abstract

On the maximum number of colors for proper anti-rainbow colorings on a planar quadrangulation, an upper bound was given by Enami-Ozeki-Yamaguchi in terms of the independence number. In this paper, as an extension, we introduce the half-monochromatic coloring on a plane graph with even polygonal faces, and give an upper bound on the maximum number of colors for such colorings in terms of the independence number.

The half-monochromatic colorings of plane graphs with even polygonal faces

TL;DR

The paper addresses finding the maximum number of colors in half-monochromatic colorings of plane graphs with even polygonal faces, extending the known result for plane quadrangulations. It develops a method based on dividing systems in the medial graph and the division tree to connect color capacity to the independence number and proves the bound . A corollary shows , where is the maximum number of regions determined by a dividing system, and the construction attains equality in cases achieving the bound. The approach also confirms that a half-monochromatic coloring exists for all such graphs and reinforces the link between planar coloring constraints and classical graph invariants.

Abstract

On the maximum number of colors for proper anti-rainbow colorings on a planar quadrangulation, an upper bound was given by Enami-Ozeki-Yamaguchi in terms of the independence number. In this paper, as an extension, we introduce the half-monochromatic coloring on a plane graph with even polygonal faces, and give an upper bound on the maximum number of colors for such colorings in terms of the independence number.
Paper Structure (2 sections, 2 theorems, 16 equations, 3 figures)

This paper contains 2 sections, 2 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['1.1']}

Key Result

Theorem 1.1

Let $G$ be a plane graph with even polygonal faces. Let $\chi_f(G)$ be the maximum integer $k$ such that $G$ admits a half-monochromatic $k$-coloring, and let $\alpha(G)$ be the independence number of $G$. Then $\chi_f(G) \le \frac{3}{2}\alpha(G)$ holds.

Figures (3)

  • Figure 1: A dividng system $\lambda$ of $G$
  • Figure 2: The division tree $T_\Lambda$ of a dividing system $\Lambda$
  • Figure 3: An example of $\Lambda$

Theorems & Definitions (10)

  • Theorem 1.1
  • proof : Proof of Theorem \ref{['1.1']}
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Claim 3
  • proof
  • Corollary 2.1
  • proof