Minimal Regular graphs with every edge in a triangle
Abstract
Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For rregular 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 rregular graphs with r >= 6 are also given. In the 5regular 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.05879
 Bibcode:
 2021arXiv210605879P
 Keywords:

 Mathematics  Combinatorics;
 05C75;
 05C35
 EPrint:
 9 pages, 6 figures