Constructing Triangle Decomposable Multigraphs with Minimum Multi-edges

TL;DR

This paper investigates methods to construct triangle decomposable multigraphs by adding minimal multi-edges, focusing on planar and toroidal graphs, to ensure all edges are part of triangles.

Contribution

It introduces new constructions for strongly triangle divisible graphs with minimal multi-edge additions, expanding understanding of triangle decompositions in complex graph classes.

Findings

Identifies classes of planar graphs that can be made triangle decomposable with minimal multi-edges.

Extends the concept to a class of toroidal graphs, demonstrating broader applicability.

Provides construction techniques for strongly triangle divisible graphs.

Abstract

We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly triangle divisible graph that is also triangle decomposable. We explore several classes of planar graphs as well as a class of toroidal graphs.