The complexity of colouring symmetric relational systems

Abstract

A relational system, S, is a set T with relations R 1, R 2, …, R k on T, denoted S = ( T, R 1, R 2, …, R k ). We consider relational systems where all the relations are binary, symmetric and antireflexive. The underlying graph, G, of a relational systems S = ( T, R 1, …, R k ) is the graph with vertex set V( G) = T and uv ∈( G) if ( u, v) ∈ R i for some i. We use the terms path (respectively) tree) to mean a relational system whose underlying graph is a path (respectively tree). Let E i ( G) = { uv|( u, v) ∈ R i } be the set of edges coloured i. The relational system defined by ( T, R 1, R 2, …, R k ) can be represented in the form of a coloured graph V( G) together with E 1( G), E 2( G), …, E k ( G). A homomorphism G → H is a mapping to V( G) to V( H) that preserves edge colours. Such a mapping is called an H-colouring of G. For a fixed H, the H-colouring problem is: H - COL ( H-colouring). Instance: A relational system G. Question: Does there exist an H-colouring of G? We show that H-COL is polynomial when H is a path and we show that trees exist such that H-COL is NP-complete. These results parallel the results of directed graphs concerning oriented paths and trees. However, the trees presented here are somewhat smaller and simpler than the known examples for directed graphs.