Intersection patterns of subtree families and colorful fractional Helly theorems

Abstract

In this paper, we study families of subtrees on a tree. Our first result gives a geometrical interpretation of a tree decomposition. We introduce acyclic families of convex bodies in $\mathds{R}^2$, which we call $\alpha$-decompositions, whose intersection graph contains a given graph as a subgraph. We then define an $\alpha$-dimension, which is the minimum value of the dimension of the nerve complex among all $\alpha$-decomposition for a graph. We prove that for a finite simple undirected graph, it is an intersection graph of a subtree family if and only if it is an intersection graph of an acyclic family of convex bodies in $\mathds{R}^2$. This implies that $\alpha$-dimension of a graph is same as the tree-width of the graph. The second result is about the intersection patterns of subtree families. Helly`s theorem \cite{Hel23} states that if $\mathcal{F}$ is a finite family of convex sets in $\mathds{R}^d$ such that every $(d+1)$-tuple of $\mathcal{F}$ is intersecting, then the whole family $\mathcal{F}$ is intersecting. It is well-known that Helly`s theorem also holds for subtree families. We first prove that, for a subtree family $\mathcal{T}$ with two color classes, if every colorful pair of $\mathcal{T}$ is intersecting, then some color class is intersecting. This is a colorful version of Helly`s theorem for subtree families. We also show that, for every $0<\alpha\leq1$, there exists $0<\beta=\beta(\alpha)\leq1$ such that, for a subtree family $\mathcal{T}$, if only an $\alpha$ fraction of pairs of $\mathcal{T}$ are intersecting, then $\mathcal{T}$ has an intersecting subfamily containing a $\beta$ fraction of the subtrees in $\mathcal{T}$. This is a fractional version of Helly`s theorem for subtree families. Finally, we prove a colorful version of fractional Helly theorem. That is, for every $0<\alpha\leq1$, there exists $0<\beta=\beta(\alpha)\leq1$ such that, for a subtree family $\mathcal{T}$ with two color classes, if only an $\alpha$ fraction of th...