On the number of connected sets in bounded degree graphs

Kustaa Kangas, Petteri Kaski, Janne H. Korhonen, Mikko Koivisto

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Abstract

A set of vertices in a graph is connected if it induces a connected subgraph. Using Shearer’s entropy lemma and a computer search, we show that the number of connected sets in a graph with n vertices and maximum vertex degree d is at most O(1.9351 n ) for d = 3, O(1.9812 n ) for d = 4, and O(1.9940 n ) for d = 5. Dually, we construct infinite families of graphs where the number of connected sets is at least Ω(1.7651 n ) for d = 3, Ω(1.8925 n ) for d = 4, and Ω(1.9375 n ) for d = 5.

Original languageEnglish
Article number#P4.34
Pages (from-to)1-19
JournalElectronic Journal of Combinatorics
Volume25
Issue number4
DOIs
Publication statusPublished - 1 Jan 2018
MoE publication typeA1 Journal article-refereed

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