Positive Co‐Degree Density of Hypergraphs

Halfpap, Anastasia; Lemons, Nathan Wishard; Palmer, Cory

doi:10.1002/jgt.23260

Positive Co‐Degree Density of Hypergraphs

Journal Article · Wed Jun 11 00:00:00 EDT 2025 · Journal of Graph Theory

DOI:https://doi.org/10.1002/jgt.23260· OSTI ID:2572547

Halfpap, Anastasia ^[1]; ^[2]; ^[3]

Iowa State Univ., Ames, IA (United States)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Univ. of Montana, Missoula, MT (United States)

The minimum positive co-degree of a nonempty r-graph H, denoted $$δ^+_{r-1}$$ (H), is the maximum k such that if S is an (r-1)-set contained in a hyperedge of H, then is contained in at least distinct hyperedges of H. Given an r-graph F, we introduce the positive co-degree Turán number co⁺ex(n, F) as the maximum positive co-degree $$δ^+_{r-1}$$ (H) over all n-vertex r-graphs H that do not contain F as a subhypergraph. In this paper, we concentrate on the behavior of co⁺ex(n, F) for 3-graphs F. In particular, we determine asymptotics and bounds for several well-known concrete 3-graphs F (e.g. $$K^-_4$$ and the Fano plane). Here, we also show that, for r-graphs, the limit γ⁺ (F)≔ lim$$_{n→∞}$$ $$\frac{co^+ex(n, F)}{n}$$ exists, and “jumps” from 0 to 1/r, that is, it never takes on values in the interval . Moreover, we characterize which r-graphs F have γ⁺ (F) = 0. Our motivation comes primarily from the study of (ordinary) co-degree Turán numbers where a number of results have been proved that inspire our results.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: Simons Foundation; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001

OSTI ID:: 2572547

Report Number(s):: LA-UR--23-30585; 1097-0118; 10.1002/jgt.23260

Journal Information:: Journal of Graph Theory, Journal Name: Journal of Graph Theory Journal Issue: 2 Vol. 110; ISSN 0364-9024; ISSN 1097-0118

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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