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arXiv · 1308.0081

Quartic graphs with every edge in a triangle

Florian Pfender·Gordon F. Royle
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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}.

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Florian Pfender, Gordon F. Royle. 2013-08-01. Quartic graphs with every edge in a triangle. https://arxiv.org/abs/1308.0081

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