Graphs and Homomorphisms

Aside from ordered sets, the fixed point property has been investigated in other settings. On one hand, the fixed point property is most likely originated in topology. (See Exercise 6-1 for the topological fixed point property.) On the other hand, in any branch of mathematics in which the underlying structures have a natural type of morphism, we can define a fixed point property as “every endomorphism has a fixed point.” Hence, graphs are natural discrete structures for which to investigate the fixed point property. However, we must be careful: For graphs, the natural analogues of order-preserving maps would be the simplicial homomorphisms from Definition 6.1. Nonetheless, graph theorists prefer the notion of a (graph) homomorphism from Definition 6.10. It would be fruitless to argue whether simplicial homomorphisms or graph homomorphisms are “more natural.” Each notion is natural in certain settings: For example, simplicial homomorphisms are in one-to-one correspondence with the simplicial maps of the clique complex of a graph, as we will see in Exercise 9-19 when we investigate simplicial complexes, whereas we will see in this chapter that graph homomorphisms are a good framework to investigate natural graph theoretical topics, such as, for example, colorings.