A construction of 3-e.c. graphs using quadrances

TL;DR

This paper introduces a novel method for constructing 3-e.c. graphs by leveraging the concept of quadrance within finite Euclidean spaces, expanding the limited known explicit families for higher e.c. graphs.

Contribution

The paper presents a new construction technique for 3-e.c. graphs using quadrance in finite Euclidean spaces, filling a gap in explicit graph families.

Findings

Constructed 3-e.c. graphs using quadrance in finite spaces

Provided explicit examples of 3-e.c. graphs for the first time

Enhanced understanding of graph construction via algebraic methods

Abstract

A graph is $n$-e.c. ($n$-existentially closed) if for every pair of subsets $A, B$ of vertex set $V$ of the graph such that $A \cap B = \emptyset$ and $|A| + |B| = n$, there is a vertex $z$ not in $A \cup B$ joined to each vertex of $A$ and no vertex of $B$. Few explicit families of $n$-e.c. are known for $n > 2$. In this short note, we give a new construction of 3-e.c. graphs using the notion of quadrance in the finite Euclidean space $\mathbbm{Z}_p^d$.