Decomposing almost complete graphs by random trees

TL;DR

This paper proves that almost surely, random trees with a given number of edges can decompose certain large, almost complete graphs, advancing understanding of graph decompositions related to Ringel's conjecture.

Contribution

It demonstrates that random trees asymptotically almost surely decompose large, almost complete graphs, extending known bounds and providing new insights into graph decomposition problems.

Findings

Random trees with m edges decompose certain large graphs asymptotically almost surely.

Decomposition results hold for graphs obtained by replacing vertices with cocliques.

A random tree with m+1 edges decomposes nearly complete graphs minus one edge.

Abstract

An old conjecture of Ringel states that every tree with $m$ edges decomposes the complete graph $K_{2m+1}$. The best known lower bound for the order of a complete graph which admits a decomposition by every given tree with $m$ edges is $O(m^3)$. We show that asymptotically almost surely a random tree with $m$ edges and $p=2m+1$ a prime decomposes $K_{2m+1}(r)$ for every $r\ge 2$, the graph obtained from the complete graph $K_{2m+1}$ by replacing each vertex by a coclique of order $r$. Based on this result we show, among other results, that a random tree with $m+1$ edges a.a.s. decomposes the compete graph $K_{6m+5}$ minus one edge.