Star decompositions via orientations

TL;DR

This paper proves that random regular graphs can be decomposed into $k$-stars with specific distributions of centers, extending previous results and providing a more general, self-contained proof for a wider range of parameters.

Contribution

It offers a direct, self-contained proof for $k$-star decompositions in random regular graphs for all $d$ and $k<d/2-1$, with stronger results and prescribed vertex sets.

Findings

Random $d$-regular graphs asymptotically almost surely have $k$-star decompositions.

The number of stars centered at each vertex is either $s$ or $s+1$.

Prescribed sets of vertices with $s$ stars are possible under certain conditions.

Abstract

A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: given $k \leq d/2$, does the random $d$-regular graph have a $k$-star decomposition (asymptotically almost surely, provided that the number of edges is divisible by $k$)? Delcourt, Greenhill, Isaev, Lidick\'y, and Postle proved the a.a.s. existence for every odd $k$ using earlier results regarding orientations satisfying certain degree conditions modulo $k$. In this paper we give a direct, self-contained proof that works for every $d$ and every $k<d/2-1$. In fact, we prove stronger results. Let $s\geq 1$ denote the integer part of $d/(2k)$. We show that the random $d$-regular graph a.a.s. has a $k$-star decomposition such that the number of stars centered at each vertex is either $s$ or $s+1$. Moreover, if $k < d/3$ or $k \leq d/2 - 2.6 \log d$, we can even prescribe the set of vertices with $s$ stars, as long as it is of the appropriate size.