Minimal Regular graphs with every edge in a triangle
James Preen

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.
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Taxonomy
TopicsAdvanced Graph Theory Research · graph theory and CDMA systems · Finite Group Theory Research
