Mei-hui CHENG, Zi-hong TIAN
A hybrid triple system of order v and index λ, denoted by HTS(v, λ), is a pair (X, B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X, such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v,λ), denoted by OLHTS(v,λ), is a collection {(Y \{y},Ai)}i, such that Y is a (v + 1)-set, each (Y \{y},Ai) is an HTS(v,λ) and all Ais form a partition of all cyclic triples and transitive triples on Y. In this paper, we shall discuss the existence problem of OLHTS(v,λ) and give the following conclusion: there exists an OLHTS(v,λ) if and only if λ= 1, 2, 4, v≡0, 1 (mod 3) and v≥ 4.
h(e) is the fractional degree of x in G. Set Eh = {e : e ∈ E(G) and h(e) ≠ 0}. If Gh is a spanning subgraph of G such that E(Gh) = Eh, then Gh is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph Gn,p with p ≥ n-2/3, almost surely Gn,p contains an fractional f-factor.
