Although any isomorphism between two graphs is a homomorphism, the study of homomorphisms between graphs has quite a different flavour to the study of isomorphisms. In this chapter we support this claim by introducing a number of topics involving graph homomorphisms. We consider the relationship between homomorphisms and graph products, and in particular a famous unsolved conjecture of Hedetniemi, which asserts that if two graphs are not n-colourable, then neither is their product. Our second major topic is the exploration of the core of a graph, which is the minimal subgraph of a graph that is also a homomorphic image of the graph. Studying graphs that are equal to their core leads us to an interesting class of graphs first studied by Andrásfai. We finish the chapter with an exploration of the cores of vertex-transitive graphs.
KeywordsConnected Graph Chromatic Number Cayley Graph Common Neighbour Petersen Graph
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- A. E. Brouwer, Finite graphs in which the point neighbourhoods are the maximal independent sets, in From universal morphisms to megabytes: a Baayen space Odyssey, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1994, 231–233.Google Scholar
- R. Häggkvist, Odd cycles of specified length in nonbipartite graphs, in Graph Theory, North-Holland, Amsterdam, 1982, 89–99.Google Scholar
- G. Hahn and C. Tardif, Graph homomorphisms: structure and symmetry, in Graph symmetry (Montreal, PQ, 1996), Kluwer Acad. Publ., Dordrecht, 1997, 107–166.Google Scholar
- W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley, 2000.Google Scholar