Circular colouring and algebraic no-homomorphism theorems
European Journal of Combinatorics
Bounds for visual cryptography schemes
Discrete Applied Mathematics
On the isoperimetric spectrum of graphs and its approximations
Journal of Combinatorial Theory Series B
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In this paper we introduce some general necessary conditions for the existence of graph homomorphisms, which hold in both directed and undirected cases. Our method is a combination of Diaconis and Saloff–Coste comparison technique for Markov chains and a generalization of Haemers interlacing theorem. As some applications, we obtain a necessary condition for the spanning subgraph problem, which also provides a generalization of a theorem of Mohar (1992) as a necessary condition for Hamiltonicity. In particular, in the case that the range is a Cayley graph or an edge-transitive graph, we obtain theorems with a corollary about the existence of homomorphisms to cycles. This, specially, provides a proof of the fact that the Coxeter graph is a core. Also, we obtain some information about the cores of vertex-transitive graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 15–38, 2003