### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 1593-1599 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 339 |

Issue number | 5 |

DOIs | |

State | Published - 6 May 2016 |

### Fingerprint

### Keywords

- Dominating sets
- Extremal enumeration
- Shearer's lemma

### Cite this

*Discrete Mathematics*,

*339*(5), 1593-1599. https://doi.org/10.1016/j.disc.2015.12.011

}

*Discrete Mathematics*, vol. 339, no. 5, pp. 1593-1599. https://doi.org/10.1016/j.disc.2015.12.011

**Counting dominating sets and related structures in graphs.** / Cutler, Jonathan; J. Radcliffe, A.

Research output: Contribution to journal › Article › Research › peer-review

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

VL - 339

SP - 1593

EP - 1599

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 5

ER -