Minimalist designs

TL;DR

This paper demonstrates the effectiveness of the iterative absorption method in graph decompositions, providing simplified proofs for triangle decompositions in large minimum degree and quasi-random graphs, advancing combinatorial design theory.

Contribution

It offers a simplified proof technique for triangle decompositions, extending results to quasi-random graphs, and illustrates the power of the iterative absorption method.

Findings

Large minimum degree triangle-divisible graphs have triangle decompositions

Quasi-random graphs also admit triangle decompositions

Simplified proof approach enhances understanding of graph decompositions

Abstract

The iterative absorption method has recently led to major progress in the area of (hyper-)graph decompositions. Amongst other results, a new proof of the Existence conjecture for combinatorial designs, and some generalizations, was obtained. Here, we illustrate the method by investigating triangle decompositions: we give a simple proof that a triangle-divisible graph of large minimum degree has a triangle decomposition and prove a similar result for quasi-random host graphs.