A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

TL;DR

This paper extends an information-theoretic method to bound the number of independent sets in irregular bipartite graphs, confirming a conjectured bound and broadening the applicability of previous results.

Contribution

It generalizes Kahn's entropy-based proof technique to irregular bipartite graphs, validating a conjectured upper bound in this broader setting.

Findings

Extended entropy-based proof technique to irregular bipartite graphs.

Confirmed the conjectured upper bound for graphs with one regular bipartite side.

Unified the bounds for regular and irregular bipartite graphs.

Abstract

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn's information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but it may be irregular in the other, the extended entropy-based proof technique yields the same bound that was conjectured by Kahn (2001) and proved by Sah et al. (2019).