On chromatic vertex stability of 3-chromatic graphs with maximum degree 4

TL;DR

This paper constructs infinite families of 3-chromatic graphs with maximum degree 4 that demonstrate specific chromatic vertex stability properties, providing insights into the stability of graph colorings.

Contribution

It presents the first known infinite families of graphs with maximum degree 4 and specific stability parameters, partially answering a previously posed problem.

Findings

Constructed graphs with $ ext{max degree}=4$, $ ext{chromatic number}=3$, $ ext{vertex stability}=3$, and independent vertex stability=2.

Provided a partial negative answer to an open problem about chromatic vertex stability.

Expanded understanding of stability properties in 3-chromatic graphs with bounded degree.

Abstract

The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $\chi(G-S)=\chi(G)-1$. In this paper we construct infinitely many graphs $G$ with $\Delta(G)=4$, $\chi(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.