Nontrivial independent sets of bipartite graphs and cross-intersecting families

TL;DR

This paper characterizes the size and structure of maximal nontrivial independent sets in certain bipartite graphs and applies these findings to bounds on cross-intersecting families in combinatorics.

Contribution

It provides a formula for the size of maximal nontrivial independent sets in bipartite graphs with transitive automorphism groups and describes their structures.

Findings

Maximal nontrivial independent set size: |Y|-d(X)+1

Structural description of these sets

Upper bounds for cross-intersecting families

Abstract

Let $G(X,Y)$ be a connected, non-complete bipartite graph with $|X|\leq |Y|$. An independent set $A$ of $G(X,Y)$ is said to be trivial if $A\subseteq X$ or $A\subseteq Y$. Otherwise, $A$ is nontrivial. By $\alpha(X,Y)$ we denote the size of maximal-sized nontrivial independent sets of $G(X,Y)$. We prove that if the automorphism group of $G(X,Y)$ is transitive on $X$ and $Y$, then $\alpha(X,Y)=|Y|-d(X)+1$, where $d(X)$ is the common degree of vertices in $X$. We also give the structures of maximal-sized nontrivial independent sets of $G(X,Y)$. As applications of this result, we give the upper bound of sizes of two cross-$t$-intersecting families of finite sets, finite vector spaces and permutations.