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Difference of facial achromatic numbers between two triangular embeddings of a graph

Kengo Enami, Yumiko Ohno

Abstract

A facial $3$-complete $k$-coloring of a triangulation $G$ on a surface is a vertex $k$-coloring such that every triple of $k$-colors appears on the boundary of some face of $G$. The facial $3$-achromatic number $ψ_3(G)$ of $G$ is the maximum integer $k$ such that $G$ has a facial $3$-complete $k$-coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge. For two triangulations $G$ and $G'$ on a surface, $ψ_3(G)$ may not be equal to $ψ_3(G')$ even if $G$ is isomorphic to $G'$ as graphs. Hence, it would be interesting to see how large the difference between $ψ_3(G)$ and $ψ_3(G')$ can be. We shall show that the upper bound for such difference in terms of the genus of the surface.

Difference of facial achromatic numbers between two triangular embeddings of a graph

Abstract

A facial -complete -coloring of a triangulation on a surface is a vertex -coloring such that every triple of -colors appears on the boundary of some face of . The facial -achromatic number of is the maximum integer such that has a facial -complete -coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge. For two triangulations and on a surface, may not be equal to even if is isomorphic to as graphs. Hence, it would be interesting to see how large the difference between and can be. We shall show that the upper bound for such difference in terms of the genus of the surface.

Paper Structure

This paper contains 5 sections, 5 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related results
  3. Cycles in a triangulation
  4. Proof of Theorem \ref{['upper']}
  5. Facial complete colorings of non-simple graphs

Key Result

Theorem 1

Let $G$ be a graph which has two triangulations $f_1(G)$ and $f_2(G)$ on a surface $\mathbb{F}$, and let $g$ be the Euler genus of $\mathbb{F}$. If $\mathbb{F}$ is orientable, then If $\mathbb{F}$ is non-orientable, then

Figures (1)

  • Figure 1: Two embeddings of $G$ on the sphere.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4: Malnič and Mohar MM
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['upper']}
  • Claim 6
  • ...and 1 more