Turán-Type Problems on a, b-Factors of Graphs, and Beyond

Given a set of graphs H, we say that a graph G is H-free if it does not contain any member of H as a subgraph. Let ex (n, H) (resp. exₒ₏ (n, H) ) denote the maximum size (resp. spectral radius) of an n-vertex H-free graph. Denote by Ex (n, H) the set of all n-vertex H-free graphs with ex (n, H) edges. Similarly, let Exₒ₏ (n, H) be the set of all n-vertex H-free graphs with spectral radius exₒ₏ (n, H). For positive integers a, b with a b, an a, b-factor of a graph G is a spanning subgraph F of G such that a dF (v) b for all v V (G), where dF (v) denotes the degree of the vertex v in F. Let F₀, ₁ be the set of all the a, b-factors of an n-vertex complete graph Kₙ. In this paper, we determine the Tur\'an number ex (n, F₀, ₁) and the spectral Tur\'an number exₒ₏ (n, F₀, ₁), respectively. Furthermore, the bipartite analogue of ex (n, F₀, ₁) (resp. exₒ₏ (n, F₀, ₁) ) is also obtained. All the corresponding extremal graphs are identified. Consequently, one sees that Exₒ₏ (n, F₀, ₁) Ex (n, F₀, ₁) holds for graphs and bipartite graphs. This partially answers an open problem proposed by Liu and Ning LN2023. Our results may deduce a main result of Fan and Lin FL2022.