Graphs with large minimum degree and no small odd cycles are $3$-colourable

Julia Böttcher; Nóra Frankl; Domenico Mergoni Cecchelli; Olaf Parczyk; Jozef Skokan

Graphs with large minimum degree and no small odd cycles are $3$-colourable

Combinatorics 2023-03-08 v2

Authors: Julia Böttcher , Nóra Frankl , Domenico Mergoni Cecchelli , Olaf Parczyk , Jozef Skokan

Abstract

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$ -vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$ -colourable. In fact, we prove a stronger result that works with a slightly smaller minimum degree.

Keywords

graph coloring graph cycles hadwiger conjecture

Cite

@article{arxiv.2302.01875,
  title  = {Graphs with large minimum degree and no small odd cycles are $3$-colourable},
  author = {Julia Böttcher and Nóra Frankl and Domenico Mergoni Cecchelli and Olaf Parczyk and Jozef Skokan},
  journal= {arXiv preprint arXiv:2302.01875},
  year   = {2023}
}

Related papers

View all related →