On the chromatic uniqueness of $K_4$-homeomorphs with girth 7

TL;DR

This paper resolves an open question from 2008 regarding the chromatic uniqueness of $K_4$-homeomorphs with girth 7, completing the classification of their chromatic properties.

Contribution

It provides a complete characterization of the chromatic uniqueness of $K_4$-homeomorphs with girth 7, addressing a long-standing open problem.

Findings

Established the chromatic uniqueness for all $K_4$-homeomorphs with girth 7

Closed the gap in the classification of $K_4$-homeomorphs' chromatic properties

Resolved the open problem from the 2008 Discrete Mathematics article

Abstract

This paper settled the question, which remains open in article published in Discrete Mathematics in 2008. Graphs homeomorphic to $K_4$, i.e., cliques on 4 vertices with edges replaced by paths are considered in this work. This work completes the study of chromaticity of $K_4$-homeomorphs of girth 7.