Intersecting Families of Spanning Trees

TL;DR

This paper establishes a combinatorial bound on the size of intersecting families of spanning trees in complete graphs, extending Erdős–Ko–Rado type results using advanced probabilistic techniques.

Contribution

It proves an Erdős–Ko–Rado type theorem for t-intersecting spanning trees of complete graphs, identifying the structure of maximal families for large n.

Findings

Largest t-intersecting families contain all trees with a fixed set of t disjoint edges

Existence of a constant C such that for n ≥ C(log n)t, the structure of maximal families is characterized

Uses spread approximation and Lovász Local Lemma techniques

Abstract

A family $\mathcal{F}$ of spanning trees of the complete graph on $n$ vertices $K_n$ is \emph{$t$-intersecting} if any two members have a forest on $t$ edges in common. We prove an Erd\H{o}s--Ko--Rado result for $t$-intersecting families of spanning trees of $K_n$. In particular, we show there exists a constant $C > 0$ such that for all $n \geq C (\log n) t$ the largest $t$-intersecting families are the families consisting of all trees that contain a fixed set of $t$ disjoint edges (as well as the stars on $n$ vertices for $t = 1$). The proof uses the spread approximation technique in conjunction with the Lopsided Lov\'asz Local Lemma.