Random 4-regular graphs have 3-star decompositions asymptotically almost surely

TL;DR

This paper proves that random 4-regular graphs almost surely have a 3-star decomposition when the number of vertices is divisible by 3, confirming a conjecture for typical cases.

Contribution

It establishes that almost all 4-regular graphs admit a 3-star decomposition, using probabilistic methods, despite counterexamples in specific classes.

Findings

Random 4-regular graphs have 3-star decompositions asymptotically almost surely.

The result holds when the number of vertices is divisible by 3.

The proof uses the small subgraph conditioning method.

Abstract

In 2006, Barat and Thomassen conjectured in 2006 that the edges of every planar 4-regular 4-edge-connected graph can be decomposed into copies of the star with 3 leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, we prove that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided the number of vertices is divisible by 3.