A note on cycle lengths in graphs of chromatic number five and six

TL;DR

This paper proves that non-complete critical graphs with chromatic numbers 5 and 6 contain cycles of all lengths modulo their chromatic number, advancing understanding of cycle structures in such graphs.

Contribution

It establishes that all non-complete (k+1)-critical graphs for k=4,5 have cycles of every length modulo k, a new result in graph cycle theory.

Findings

Non-complete 5-critical graphs contain cycles of all lengths modulo 4.

Non-complete 6-critical graphs contain cycles of all lengths modulo 5.

Abstract

In this note, we prove that every non-complete $(k+1)$-critical graph contains cycles of all lengths modulo $k$, where $k=4,5$.