Some lower bounds for the $L$-intersection number of graphs

TL;DR

This paper establishes lower bounds on the $L$-intersection number and bipartite $L$-intersection number of graphs, relating these bounds to the minimum rank of the graphs, advancing understanding of intersection representations.

Contribution

It introduces new lower bounds for the $L$-intersection number based on the minimum rank of graphs, applicable to various sets $L$, and extends these bounds to bipartite cases.

Findings

Lower bounds for $L$-intersection number in terms of minimum rank.

Extension of bounds to bipartite $L$-intersection number.

Applicable to various types of set $L$.

Abstract

For a set of non-negative integers $L$, the $L$-intersection number of a graph is the smallest number $l$ for which there is an assignment on the vertices to subsets $A_v \subseteq \{1,\dots, l\}$, such that every two vertices $u,v$ are adjacent if and only if $|A_u \cap A_v|\in L$. The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this paper, some lower bounds for the (bipartite) $L$-intersection number of a graph for various types $L$ in terms of the minimum rank of graph are obtained.