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Kempe Equivalent List Colorings Revisited

Dibyayan Chakraborty, Carl Feghali, Reem Mahmoud

Abstract

A \emph{Kempe chain} on colors $a$ and $b$ is a component of the subgraph induced by colors $a$ and $b$. A \emph{Kempe change} is the operation of interchanging the colors of some Kempe chain. For a list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe change is \emph{$L$-valid} for $\varphi$ if performing the Kempe change yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from the other by a sequence of $L$-valid Kempe changes. A \emph{degree-assignment} is a list-assignment $L$ such that $L(v)\ge d(v)$ for every $v\in V(G)$. Cranston and Mahmoud (\emph{Combinatorica}, 2023) asked: For which graphs $G$ and degree-assignment $L$ of $G$ is it true that all the $L$-colorings of $G$ are $L$-equivalent? We prove that for every 4-connected graph $G$ which is not complete and every degree-assignment $L$ of $G$, all $L$-colorings of $G$ are $L$-equivalent.

Kempe Equivalent List Colorings Revisited

Abstract

A \emph{Kempe chain} on colors and is a component of the subgraph induced by colors and . A \emph{Kempe change} is the operation of interchanging the colors of some Kempe chain. For a list-assignment and an -coloring , a Kempe change is \emph{-valid} for if performing the Kempe change yields another -coloring. Two -colorings are \emph{-equivalent} if we can form one from the other by a sequence of -valid Kempe changes. A \emph{degree-assignment} is a list-assignment such that for every . Cranston and Mahmoud (\emph{Combinatorica}, 2023) asked: For which graphs and degree-assignment of is it true that all the -colorings of are -equivalent? We prove that for every 4-connected graph which is not complete and every degree-assignment of , all -colorings of are -equivalent.
Paper Structure (4 sections, 6 theorems, 3 equations, 2 figures)

This paper contains 4 sections, 6 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Key lemmas
  3. Outline of the proof
  4. The proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1

Let $G$ be a connected graph and $L$ be a list-assignment for $G$. If $|L(x)|=\Delta(G)$ for every $x\in V(G)$ and $\Delta(G)\ge3$, then all $L$-colorings of $G$ are $L$-equivalent unless $L(v) = L(u)$ for every $u, v\in V(G)$ and $G$ is isomorphic to the complete graph or the triangular prism.

Figures (2)

  • Figure 1: $N^2(\varphi(v),a)\neq\emptyset$, so there exists $z$ with $a\notin L(z)$.
  • Figure 2: (Left) Case 1: $t\in N^{3^+}(\phi(x),c)$ and $\beta\notin L(t)$. (Right) Case 2: $t\in N^2(\phi(x),c)$, so $c$ is special for $z$.

Theorems & Definitions (18)

  • Theorem 1: cranston2021kempe
  • Theorem 2
  • Conjecture 1
  • Lemma 1: gallai1963kritischeborodin1979problemserdos1979choosability
  • Lemma 2
  • Lemma 3: cranston2021kempelas1981kempe
  • Lemma 4: cranston2021kempe
  • proof
  • proof : Proof of Lemma \ref{['lem:equiv2']}
  • proof : Proof of Theorem \ref{['thm:main2']}
  • ...and 8 more