On number of different sized induced subgraphs of Bipartite-Ramsey graphs

TL;DR

This paper studies the sizes of induced subgraphs in bipartite graphs, introduces $C$-$Bipartite$-$Ramsey$ graphs, and provides evidence supporting the conjecture that $K_{n,n}$ minimizes the multiplication table among bipartite graphs.

Contribution

It defines $C$-$Bipartite$-$Ramsey$ graphs and proves they typically have large multiplication tables, supporting a conjecture about minimality of $K_{n,n}$.

Findings

Most $C$-$Bipartite$-$Ramsey$ graphs have multiplication tables of size proportional to the number of edges.

Provides evidence that $K_{n,n}$ minimizes the multiplication table among bipartite graphs with $n^2$ edges.

Abstract

In this paper, we investigate the set of sizes of induced subgraphs of bipartite graphs. We introduce the definition of $C$-$Bipartite$-$Ramsey$ graphs, which is closely related to Ramsey graphs and prove that in `most' cases, these graphs have multiplication tables of $\Omega(e(G))$ in size. We apply our result to give direct evidence to the conjecture that the complete bipartite graph $K_{n,n}$ is the minimiser of the multiplication table on $n^2$ edges raised by Narayanan, Sahasrabudhe and Tomon.