Hedetniemi's conjecture and strongly multiplicative graphs

TL;DR

This paper explores strongly multiplicative graphs related to Hedetniemi's conjecture, introduces new classes of such graphs, and revisits existing proofs with combinatorial methods to understand the boundaries of current techniques.

Contribution

It defines strongly multiplicative graphs, provides new examples of such graphs, and offers a different combinatorial approach to existing proofs, aiming to understand the limits of current methods.

Findings

All graphs where every edge is in at most a square are strongly multiplicative.

The third power of any graph with girth >12 is strongly multiplicative.

Current methods do not progress on cliques, indicating potential limits of these techniques.

Abstract

A graph K is multiplicative if a homomorphism from any product G x H to K implies a homomorphism from G or from H. Hedetniemi's conjecture states that all cliques are multiplicative. In an attempt to explore the boundaries of current methods, we investigate strongly multiplicative graphs, which we define as K such that for any connected graphs G,H with odd cycles C,C', a homomorphism from $(G \times C') \cup (C \times H) \subseteq G \times H$ to K implies a homomorphism from G or H. Strong multiplicativity of K also implies the following property, which may be of independent interest: if G is non-bipartite, H is a connected graph with a vertex h, and there is a homomorphism $\phi \colon G \times H \to K$ such that $\phi(-,h)$ is constant, then H admits a homomorphism to K. All graphs currently known to be multiplicative are strongly multiplicative. We revisit the proofs in a different view based on covering graphs and replace fragments with more combinatorial arguments. This allows us to find new (strongly) multiplicative graphs: all graphs in which every edge is in at most square, and the third power of any graph of girth >12. Though more graphs are amenable to our methods, they still make no progress for the case of cliques. Instead we hope to understand their limits, perhaps hinting at ways to further extend them.