On H-Intersecting Graph Families and Counting of Homomorphisms
Igal Sason
TL;DR
This work studies the maximal size of $\mathsf{H}$-intersecting graph families and the counting of graph homomorphisms through a primarily information-theoretic lens. Using three variants of Shearer’s inequalities, the authors derive a general bound $|\mathcal{G}| \le 2^{\binom{n}{2}- (\chi(\mathsf{H})-1)}$ for $\mathsf{H}$-intersecting families, with a relaxable bound via the Lovász $\vartheta$-function, and they connect this extremal bound to the combinatorial structure of $\mathsf{H}$ and its chromatic number. In the second part, two entropy-based methodologies yield bounds on the number of homomorphisms $\mathrm{hom}(\mathsf{T},\mathsf{G})$, including a general upper bound for perfect $\mathsf{T}$ in terms of the Lovász theta function, and nuanced analyses for complete bipartite targets $\mathsf{K}_{s,t}$ in bipartite graphs, with comparisons to Sidorenko-type lower bounds. Overall, the paper bridges extremal graph theory and information-theoretic methods, providing computable relaxations and insight into the asymptotics of homomorphism counts. The results offer both theoretical bounds and practical tools for estimating the size of intersecting graph families and for counting graph mappings in combinatorial settings.
Abstract
This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H. Such families are referred to as H-intersecting graph families. The bound is derived using the combinatorial version of Shearer's lemma, and it forms a nontrivial extension of the bound derived by Chung, Graham, Frankl, and Shearer (1986), where H is specialized to a triangle. The derived bound is expressed in terms of the chromatic number of H, while a relaxed version, formulated using the Lovász $\vartheta$-function of the complement of H, offers reduced computational complexity. Additionally, a probabilistic version of Shearer's lemma, combined with properties of the Shannon entropy, are employed to establish bounds related to the enumeration of graph homomorphisms, providing further insights into the interplay between combinatorial structures and information-theoretic principles.