Triangle Decompositions of Planar Graphs

TL;DR

This paper characterizes when planar multigraphs can be decomposed into triangles or rational combinations of triangles, providing necessary and sufficient conditions and simplifying the weights needed for such decompositions.

Contribution

It offers a complete characterization of triangle and rational triangle decompositions in planar multigraphs, including simple graphs and those with K4 as underlying graph.

Findings

Necessary and sufficient condition for triangle decomposability of planar multigraphs.

Rationally triangle decomposable simple planar graphs can be decomposed with weights 0, 1, and 1/2.

Conditions on edge multiplicities for multigraphs with K4 as underlying graph.

Abstract

A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1. We present a necessary and sufficient condition for a planar multigraph to be triangle decomposable. We also show that if a simple planar graph is rationally triangle decomposable, then it has such a decomposition using only weights 0,1 and 1/2. This result provides a characterization of rationally triangle decomposable simple planar graphs. Finally, if G is a multigraph with the complete graph of order 4 as underlying graph, we give necessary and sufficient conditions on the multiplicities of its edges for G to be triangle and rationally triangle decomposable.