The Line Graph of the Universal Homogeneous Triangle-free Graph

TL;DR

This paper investigates the structure of line graphs derived from triangle-free graphs, focusing on the universal homogeneous triangle-free graph, and explores its implications for relational complexity in structures with forbidden embeddings.

Contribution

It introduces a homogenization of the line graph of the universal homogeneous triangle-free graph within a finite relational language, advancing understanding of relational complexity.

Findings

Structural results on line graphs of triangle-free graphs

Homogenization of the line graph in a finite relational language

Implications for relational complexity of forbidden embeddings

Abstract

We present some results on the structure of line graphs of triangle-free graphs and a homogenization of the line graph of the universal homogeneous triangle-free graph in a finite relational language with arity 4. This result is relevant to the study of the relational complexity of structures defined by forbidden embeddings.