Quartic graphs with every edge in a triangle

TL;DR

This paper characterizes 4-regular multigraphs where every edge is part of a triangle, showing they are constructed from cycles, line multigraphs, or simple subgraph replacements, providing a complete structural classification.

Contribution

It provides a complete characterization of quartic multigraphs with all edges in triangles, including simple graphs, through structural descriptions and subgraph operations.

Findings

Such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or derived from these by subgraph replacements.

Simple quartic graphs with all edges in triangles are either cycle squares, line graphs of cubic graphs, or derived from line multigraphs.

The classification includes a specific construction involving replacing triangles with K_{1,1,3}.

Abstract

We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from these by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every edge in a triangle is either the square of a cycle, the line graph of a cubic graph or a graph obtained from the line multigraph of a cubic multigraph by replacing triangles with copies of K_{1,1,3}.