Existential Closure in Line Graphs

• Main Authors: Burgess, Andrea C, Luther, Robert D, Pike, David A
• Format: Journal Article
• Language: English
• Published: 02-11-2022
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.2211.01168

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.