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. exₛₚ(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 Exₛₚ(n,H) be the set of all n-vertex H-free graphs with spectral radius exₛₚ(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 Fa,b 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,Fa,b) and the spectral Tur\'an number exₛₚ(n,Fa,b), respectively. Furthermore, the bipartite analogue of ex(n,Fa,b) (resp. exₛₚ(n,Fa,b)) is also obtained. All the corresponding extremal graphs are identified. Consequently, one sees that Exₛₚ(n,Fa,b)⊆ Ex(n, Fa,b) 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}.