Kempe Classes and Almost Bipartite Graphs

Daniel W. CranstonCarl Feghali

Mar 2023

摘要

Let $G$ be a graph and $k$ be a positive integer, and let $Kc(G, k)$ denotethe number of Kempe equivalence classes for the $k$-colorings of $G$. In 2006,Mohar noted that $Kc(G, k) = 1$ if $G$ is bipartite. As a generalization, weshow that $Kc(G, k) = 1$ if $G$ is formed from a bipartite graph by adding anynumber of edges less than $\binom{\lceil k/2\rceil}2+\binom{\lfloork/2\rfloor}2$. We show that our result is tight (up to lower order terms) byconstructing, for each $k \geq 8$, a graph $G$ formed from a bipartite graph byadding $(k^2+8k-45+1)/4$ edges such that $Kc(G, k) \geq 2$. This refutes arecent conjecture of Higashitani--Matsumoto.

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