Digital Object Identifier (DOI) 10.1007/s00373-007-0712-5 Graphs and Combinatorics (2007) 23[Suppl]:275–281 Graphs and Combinatorics © Springer 2007 On the Density of Trigraph Homomorphisms Pavol Hell 1 , and Jarik Neˇ setˇ ril 2 1 School of Computing Science, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada. e-mail: pavol@cs.sfu.ca 2 KAM MFF UK, Malostransk´ en´ am 22, Praha, Czech Republic. e-mail: nesetril@kam.mff.cuni.cz Abstract. 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. Key words. Trigraphs, Homomorphisms, Density, Gaps 1. Introduction A trigraph G consists of a finite set V (G) of vertices, and two (possibly intersecting) edge sets E 1 (G) and E 2 (G) on V (G), such that E 1 (G) ∪ E 2 (G) contains all pairs of (possibly equal) vertices. Thus a trigraph G can be viewed as a superposition of two graphs on the vertex set V (G)—the graph G 1 with the edge set E 1 (G), and the graph G 2 with the edge set E 2 (G), both standard graphs (without multiple edges but) with loops allowed [7]. Alternately, we may view a trigraph as a relational structure con- sisting of a set V (G) with two symmetric binary relations E 1 (G) and E 2 (G). The only restriction we have is that each pair of (possibly equal) vertices is adjacent in at least one of the graphs G a , i.e., related in at least one of the relations E a (G), a = 1, 2. This ‘completeness’ restriction substantially affects the situation, and the questions we address here have been answered for relational structures without this restriction [7, 9]. (The ‘completeness’ restriction is in a sense similar to restricting digraphs to tournaments, cf. [8].) A trigraph G is a subtrigraph of a trigraph H if V (G) ⊆ V (H ) and E a (G) ⊆ E a (H ) for a = 1, 2. If every two vertices of G have the same relations in G as in H , we say that G is an induced subtrigraph of H . Let G and H be any trigraphs. A homomorphism f of G to H is a mapping of V (G) to V (H ) which preserves both relations, i.e., such that uv ∈ E a (G) implies f (u)f (v) ∈ E a (H ), for a = 1, 2. A bijective homomorphism of G to H is an isomorphism between G and H , and if G = H , it is an automorphism of G. Two trigraphs are homomorphically equivalent if each admits a homomorphism to the other. A trigraph is a core if it is not homomor- phically equivalent to any proper subtrigraph. Each trigraph is homomorphically