Homomorphisms of binary Cayley graphs

TL;DR

This paper investigates the structure of binary Cayley graphs, proving they contain projective cubes if non-bipartite, and explores conjectures about homomorphisms to these cubes, advancing understanding of their graph-theoretic properties.

Contribution

The paper strengthens previous results by showing non-bipartite binary Cayley graphs contain projective cubes and proposes a conjecture on homomorphisms to these cubes, with partial proofs.

Findings

Non-bipartite binary Cayley graphs contain projective cubes

Proved special cases of the homomorphism surjectivity conjecture

Extended understanding of the subgraph structure of binary Cayley graphs

Abstract

A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture.