Existential Closure in Line Graphs

TL;DR

This paper explores the properties and constructions of $n$-existentially closed line graphs, providing necessary conditions, infinite families, and specific classifications for planar and hypergraph line graphs.

Contribution

It introduces new conditions and constructions for $n$-existentially closed line graphs, including classifications for planar cases and hypergraph extensions.

Findings

Identified necessary conditions for $n$-existentially closed line graphs.

Constructed infinite families of such graphs.

Proved exactly two 2-existentially closed planar line graphs.

Abstract

A graph $G$ is {\it $n$-existentially closed} if, for all disjoint sets of vertices $A$ and $B$ with $|A\cup B|=n$, there is a vertex $z$ not in $A\cup B$ adjacent to each vertex of $A$ and to no vertex of $B$. In this paper, we investigate $n$-existentially closed line graphs. In particular, we present necessary conditions for the existence of such graphs as well as constructions for finding infinite families of such graphs. We also prove that there are exactly two $2$-existentially closed planar line graphs. We then consider the existential closure of the line graphs of hypergraphs and present constructions for $2$-existentially closed line graphs of hypergraphs.