Distinguishing graphs by their left and right homomorphism profiles

We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph. Ministerio de Educación y Ciencia MTM2008-05866-C03-01 Junta de Andalucía FQM- 0164 Junta de Andalucía P06-FQM-01649