Minimal Ramsey graphs for cyclicity

TL;DR

This paper characterizes minimal graphs ensuring every edge-coloring contains a monochromatic cycle, showing they reduce to specific base graphs and establishing related Ramsey properties and constructions.

Contribution

It identifies the structure of minimal Ramsey graphs for cyclicity, reducing them to base graphs and providing explicit constructions and applications in Ramsey theory.

Findings

Every such graph reduces to $K_5-e$ or $K_4\vee K_4$

Graphs with $e(G)\geq 2n-1$ contain these as minors

Established Ramsey infiniteness for chromatic classes 2, 3, 4

Abstract

We study graphs with the property that every edge-colouring admits a monochromatic cycle (the length of which may depend freely on the colouring) and describe those graphs that are minimal with this property. We show that every member in this class reduces recursively to one of the base graphs $K_5-e$ or $K_4\vee K_4$ (two copies of $K_4$ identified at an edge), which implies that an arbitrary $n$-vertex graph with $e(G)\geq 2n-1$ must contain one of those as a minor. We also describe three explicit constructions governing the reverse process. As an application we are able to establish Ramsey infiniteness for each of the three possible chromatic subclasses $\chi=2, 3, 4$, the unboundedness of maximum degree within the class as well as Ramsey separability of the family of cycles of length $\leq l$ from any of its proper subfamilies.