Strictly $k$-colorable graphs

TL;DR

This paper introduces the concept of strictly colorable graphs, formalizes the idea of tightness in choosability, and characterizes all strictly colorable complete multipartite graphs, advancing understanding of graph coloring constraints.

Contribution

It formalizes the notion of tightness in choosability and characterizes all strictly colorable complete multipartite graphs, expanding theoretical knowledge in graph coloring.

Findings

Characterization of all strictly colorable complete multipartite graphs

Formalization of tightness in choosability

Extension of the four color theorem to a refined scale

Abstract

Xuding Zhu introduced a refined scale of choosability in 2020 and observed that the four color theorem is tight on this scale. We formalize and explore this idea of tightness in what we call strictly colorable graphs. We then characterize all strictly colorable complete multipartite graphs.