Kempe equivalence of 4-colourings of some plane triangulations

TL;DR

This paper investigates the structure of 4-colourings in certain plane triangulations with specific degree properties, enumerates Kempe classes, and demonstrates the Kempe equivalence of colourings after removing a pole, revealing multiple new structural insights.

Contribution

It introduces the enumeration of Kempe classes for a family of plane triangulations and proves Kempe equivalence of colourings after pole removal, advancing understanding of colouring transformations.

Findings

Number of Kempe classes is at least loor(n/6)or the graph family.

All 4-colourings of H_n are Kempe equivalent up to loor(13n/2)hanges.

Enumeration of the total number of 4-colourings for G_n.

Abstract

Let $G_{n}$, where $n \geqslant 5$, be a simple plane triangulation which has $2$ non-adjacent vertices of degree $n$ (called \textit{poles} of $G_n$) and $2n$ vertices of degree~$5$. A set of Kempe equivalent $4$-colourings of $G_{n}$ is called a \textit{Kempe class}. The number of Kempe classes of $G_{n}$ is enumerated. In particular it is shown that there is at least $\lfloor \frac{n}{6} \rfloor$ Kempe classes of $G_{n}$. We say that $4$-colourings $A, B$ of $G_{n}$ are \textit{equal} if there exists a permutation~$P$ of the set of colours such that $A = P \circ B$. Otherwise, $A$, $B$ are \textit{different}. The number of different $4$-colourings of $G_{n}$ is enumerated. Suppose that $H_{n} = G_{n} - b$, where $b$ is a pole of $G_{n}$. We prove that all $4$-colourings of $H_{n}$ are Kempe equivalent up to $\lfloor \frac{13n}{2} \rfloor$ Kempe changes. %$3n$ ($\lfloor \frac{9n}{2} \rfloor$ and $\lfloor \frac{13n}{2} \rfloor$) Kempe changes, for $n \equiv 0\, (mod\, 3)$ ($n \equiv 2\, (mod\, 3)$ and $n \equiv 1\, (mod\, 3)$, respectively).