On the CNF-complexity of bipartite graphs containing no $K_{2,2}$'s

TL;DR

This paper demonstrates that certain bipartite graphs without $K_{2,2}$ subgraphs can be represented with a CNF formula using a number of clauses proportional to the logarithm of their average degree, challenging previous assumptions.

Contribution

It introduces a probabilistic construction showing bipartite graphs with no $K_{2,2}$ can have low CNF complexity, contradicting prior research expectations.

Findings

Bipartite graphs with no $K_{2,2}$ can have average degree $d$ with CNF representations using $C imes ext{log} d$ clauses.

The construction challenges existing beliefs about the CNF complexity of such graphs.

Provides a probabilistic method to analyze the CNF complexity of bipartite graphs.

Abstract

By a probabilistic construction, we find a bipartite graph having average degree $d$ which can be expressed as a conjunctive normal form using $C \log d$ clauses. This contradicts research problem 1.33 of Jukna.