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 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.05879
- arXiv:
- arXiv:2106.05879
- Bibcode:
- 2021arXiv210605879P
- Keywords:
-
- Mathematics - Combinatorics;
- 05C75;
- 05C35
- E-Print:
- 9 pages, 6 figures