Hedetniemi's Conjecture and Strongly Multiplicative Graphs

Abstract

A graph K is multiplicative if a homomorphism from any product G × 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 fromStrong 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 φ : G × H → K such that φ(−, 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.

Reconfiguring graph homomorphisms on the sphere

“…" Wrochna [21] developed ideas inspired by algebraic topology to prove the remarkably general result that H-Recolouring is solvable in polynomial time whenever H does not contain a cycle of length 4. By further refining his topological approach, Wrochna [22] (see also [23]) proved a "multiplicativity" result for graphs without cycles of length 4 which is closely connected to Hedetniemi's Conjecture [11]; very recently, Tardif and Wrochna [19] have extended these methods beyond the setting of C 4 -free graphs. On the hardness side, however, only a few examples are known.…”

Topology and Adjunction in Promise Constraint Satisfaction

“…Topological methods and adjunction (including some specific cases that we use in this paper) have also been actively used in research around Hedetniemi's conjecture about the chromatic number of graph products [34,61,68,69,70,71] (recently disproved by Shitov [67]). A few ideas in this paper are inspired by this line of research.…”