Lower Bounds on Intersection Families for Certain Graphs

TL;DR

This paper develops new lower bounds for $H$-intersecting graph families using multipartite graph decompositions and conjectures the optimality of a known bound for $P_4$, supported by computational verification.

Contribution

It introduces a general construction for $H$-intersecting families that improves existing bounds and compares favorably to prior methods, and it conjectures the optimality of a specific density bound for $P_4$.

Findings

New lower bounds for $H$-intersecting families based on graph decompositions.

Comparison showing the new bounds are wider and stronger than previous results.

Computational evidence supporting the conjecture of the optimality of the Christofides bound for $P_4$.

Abstract

A family of graphs $\mathcal{F}$ is $H$-intersecting if the edge intersection of any two graphs in $\mathcal{F}$ contains a copy of a fixed graph $H$. A fundamental problem is to determine the maximum size of such a family. The trivial lower bound of $2^{\binom{n}{2} - e(H)}$ is known to be not sharp for some graphs, such as the $P_4$ graph, as shown by Christofides. This paper presents two main contributions. First, we introduce a general construction for $H$-intersecting families based on decompositions of complete multipartite graphs, yielding new lower bounds for $H = K_{s_1, \dots, s_{k-1}, t}$. We compare this construction to a result by Balogh and Linz, showing that our bound is valid for a substantially wider range of parameters (beginning at $t \ge 2^{\sum_i s_i}$) and provides a stronger numerical bound for a large interval where both constructions are applicable. Second, we conjecture the $\frac{17}{128}$ Christofides bound for $P_4$ is optimal, which would resolve the Alon-Spencer conjecture. We computationally verify this density is optimal for families generated by connected 6-vertex host graphs with 7 or 8 edges.