Lower Bounds on Intersection Families for Certain Graphs
Paul Hamrick, Gary Hu

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 is -intersecting if the edge intersection of any two graphs in contains a copy of a fixed graph . A fundamental problem is to determine the maximum size of such a family. The trivial lower bound of is known to be not sharp for some graphs, such as the graph, as shown by Christofides. This paper presents two main contributions. First, we introduce a general construction for -intersecting families based on decompositions of complete multipartite graphs, yielding new lower bounds for . 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 ) and provides a stronger numerical bound for a large interval where both constructions are applicable. Second, we…
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Taxonomy
TopicsLimits and Structures in Graph Theory · graph theory and CDMA systems · Advanced Graph Theory Research
