Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.18064 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866918214376620032 |
|---|---|
| author | Hamrick, Paul Hu, Gary |
| author_facet | Hamrick, Paul Hu, Gary |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower Bounds on Intersection Families for Certain Graphs Hamrick, Paul Hu, Gary Combinatorics 05C35 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. |
| title | Lower Bounds on Intersection Families for Certain Graphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2510.18064 |