Minimal Regular graphs with every edge in a triangle

TL;DR

This paper investigates the structure of regular graphs where every edge is part of a triangle, establishing lower bounds on triangle counts, classifying small cases, and exploring constructions for higher degrees.

Contribution

It provides lower bounds for the number of triangles in such graphs, classifies all 5-regular cases, and introduces methods for constructing graphs with higher degrees.

Findings

Lower bounds for triangles in regular graphs with edges in triangles

Complete classification of 5-regular such graphs

Construction methods for r-regular graphs with r >= 6

Abstract

Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then classify all such graphs using line graphs and even cycle decompositions. Examples of ways to create such r-regular graphs with r >= 6 are also given. In the 5-regular case, these minimal graphs are proven to be the only regular graphs with every edge in a triangle which cannot have an edge removed and still have every edge in a triangle.