On the number of connected sets in bounded degree graphs

Abstract

A set of vertices in a graph is connected if the set induces a connected subgraph. Using Shearer’s entropy lemma, we show that the number of connected sets in an n-vertex graph with maximum vertex degree d is O(1.9351n) for d = 3, O(1.9812n) for d = 4, and O(1.9940n) for d = 5. Dually, we construct infinite families of generalized ladder graphs whose number of connected sets is bounded from below by Ω(1.5537n) for d = 3, Ω(1.6180n) for d = 4, and Ω(1.7320n) for d = 5.