Counting dominating sets and related structures in graphs

We first consider some problems related to the maximum number of dominating (or total dominating) sets in a regular graph. Our techniques, centered around Shearer’s entropy lemma, extend to a reasonably broad class of graph parameters enumerating vertex colorings that satisfy conditions on the multiset of colors appearing in neighborhoods (either open or closed). Dominating sets and total dominating sets are examples, as are graph colorings in which each vertex’s neighborhood is not monochromatic (or rainbow). In the final section, we think about a generalization of dominating sets in a slightly different direction. Just as independent sets are homomorphisms into K2K2 with one vertex looped, we think of dominating sets as an example of what we call an existence homomorphism. Here our results are substantially less complete, though we do solve some natural problems.