Erdős-Ko-Rado Theorems for Paths in Graphs
Neal Bushaw, James Danielsson, Glenn Hurlbert
TL;DR
This work extends Erdős–Ko–Rado theory to families of paths in graphs, focusing on sun graphs ${S_n^t}$ and theta graphs ${\Theta(a_1,\dots,a_k)}$ to establish 1-EKR results and, for suns, $s$-EKR results. It develops elementary transversal bounds and analyzes triangular families, culminating in sharp bounds for triangular subfamilies of projective planes ${\mathbb P}(q)$. The results connect path-intersection structure with transversals, offering Hilton–Milner-type insights in cycles and exploring broad questions about EKR phenomena in graph-path families. The work provides a framework for understanding how global graph structure governs the maximal intersecting families of paths and poses several natural extensions to broader graph classes and design-theoretic contexts.
Abstract
A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting subfamilies have size at most that of some $s$-star. For example, the classic 1961 Erdős-Ko-Rado theorem states essentially that the family of $r$-sized subsets of $\{1,2,\ldots,n\}$ is $s$-EKR when $n$ is a large enough function of $r$ and $s$, and the 1967 Hilton-Milner theorem provides the near-star structure of the largest non-star intersecting family of such sets. Two important conjectures along these lines followed: by Chvátal in 1974, that every family of sets that all subsets of its members is 1-EKR, and by Holroyd and Talbot in 2005, that, for every graph, the family of all its $r$-sized independent sets is 1-EKR when every maximal independent set has size at least $2r$. In this paper we present similar 1-EKR results for families of length-$r$ paths in graphs, specifically for sun graphs, which are cycles with pendant edges attached in a uniform way, and theta graphs, which are collections of pairwise internally disjoint paths sharing the same two endpoints. We also prove $s$-EKR results for such paths in suns, and give a Hilton-Milner type result for them as well. A set is a transversal of a family of sets if it intersects each member of the family, and the transversal number of the family is the size of its smallest transversal. For example, stars have transversal number 1, and the Hilton-Milner family has transversal number 2. We conclude the paper with some transversal results involving what we call triangular families, including a few results for projective planes.