Cycle-continuous mappings -- order structure

TL;DR

This paper investigates cycle-continuous mappings between graphs, proving the existence of graph sets with no such mappings and demonstrating that any countable poset can be represented through graphs with these mappings, advancing understanding of graph order structures.

Contribution

It introduces the concept of cycle-continuous mappings, proves the existence of graph sets with no such mappings, and shows that all countable posets can be represented via these mappings.

Findings

Existence of infinite graph sets with no cycle-continuous mappings.

Representation of any countable poset by graphs with cycle-continuous mappings.

Extension of previous conjectures and questions in graph theory.

Abstract

Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to the Petersen graph. Answering a question of DeVos, Ne\v{s}et\v{r}il, and Raspaud, we prove that there exists an infinite set of graphs with no cycle-continuous mapping between them. Further extending this result, we show that every countable poset can be represented by graphs and existence of cycle-continuous mappings between them.