Quintic graphs with every edge in a triangle

TL;DR

This paper characterizes 5-regular multigraphs where every edge is part of a triangle, identifying specific constructions and reduction processes that describe all such graphs.

Contribution

It provides a complete characterization of quintic graphs with every edge in a triangle, including new construction methods and reduction techniques.

Findings

Graphs are either small, constructed by adding perfect matchings to line graphs of cubic graphs, or reducible to these forms.

The characterization covers all such quintic graphs with the given property.

A sequence of operations can reduce complex graphs to known base cases.

Abstract

We characterise the quintic (i.e. 5-regular) multigraphs with the property that every edge lies in a triangle. Such a graph is either from a set of small graphs or is formed by adding a perfect matching to a line graph of a cubic graph as double edges, or can be reduced by a sequence of operations to one of these graphs.