TY - JOUR
T1 - Counting dominating sets and related structures in graphs
AU - Cutler, Jonathan
AU - J. Radcliffe, A.
PY - 2016/5/6
Y1 - 2016/5/6
N2 - 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 K2 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.
AB - 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 K2 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.
KW - Dominating sets
KW - Extremal enumeration
KW - Shearer's lemma
UR - http://www.scopus.com/inward/record.url?scp=84956957053&partnerID=8YFLogxK
U2 - 10.1016/j.disc.2015.12.011
DO - 10.1016/j.disc.2015.12.011
M3 - Article
AN - SCOPUS:84956957053
SN - 0012-365X
VL - 339
SP - 1593
EP - 1599
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 5
ER -