Intersecting Families of Spanning Trees of $K_{n,n}$

Abstract

A family of spanning trees of a graph is $t$-intersecting if any pair of spanning trees in the family has $t$ or more edges in common. For sufficiently large $n$ and $t \leq n/C\log_2 n$ for some absolute constant $C>0$, we give a nearly complete characterization of the extremal $t$-intersecting families of spanning trees in balanced complete bipartite graphs with parts of order $n$. In particular, for $t=1$, we give exact bounds and a full characterization of the extremal families. For $t \geq 2$, our bounds are tight up to lower-order terms, and we show that any extremal $t$-intersecting family is of the form $\mathcal{F} \cup \mathcal{S}'$ where $\mathcal{F}$ is a family of all trees containing a fixed $t$-matching, and $\mathcal{S}'$ is a distinguished set of exceptional trees of size $|\mathcal{S}'| = o(|\mathcal{F}|)$.