Searching for regular, triangle-distinct graphs

TL;DR

This paper investigates the existence of regular, triangle-distinct graphs, providing examples for orders between 21 and 27 and detailing the methods used to find these graphs.

Contribution

It presents the first known examples of regular, triangle-distinct graphs within specified orders and describes the search methodology.

Findings

Examples of regular, triangle-distinct graphs for orders 21 to 27

Methodology for constructing and identifying such graphs

Addresses an open question in graph theory

Abstract

The triangle-degree of a vertex v of a simple graph G is the number of triangles in G that contain v. A simple graph is triangle-distinct if all its vertices have distinct triangle-degrees. Berikkyzy et al. [Discrete Math. 347 (2024) 113695] recently asked whether there exists a regular graph that is triangle-distinct. Here we showcase the examples of regular, triangle-distinct graphs with orders between 21 and 27, and report on the methodology used to find them.