On the Density of Trigraph Homomorphisms

An order is dense if A < B implies A < C < B for some C. The homomorphism order of (nontrivial) graphs is known to be dense. Homomorphisms of trigraphs extend homomorphisms of graphs, and model many partitions of interest in the study of perfect graphs. We address the question of density of the homomorphism order for trigraphs. It turns out that there are gaps in the order, and we exactly characterize where they occur. ∗This work was carried out while the authors were guests of the Isaac Newton Institute for Mathematical Sciences at Cambridge, UK; the support and hospitality of the Institute is greatly appreciated.