Let n, s and t be three integers with s ≥ 1, t ≥ 0 and n = 3s + 4t. Let G be a graph of order n such that the minimum degree of G is at least (n + s)/2. Then G contains a 2-factor with s + t components such that s of them are triangles and t of them are quadrilaterals.
We prove that if G is a graph of order 5k and the minimum degree of G is at least 3k then G contains k disjoint cycles of length 5.
Let be a complete multipartite graph on with and being its binomial edge ideal. It is proved that the Castelnuovo-Mumford regularity is for any positive integer .
Let be a finite-dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules, introduced by F. Panaite, F. Van Oystaeyen (2008), is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra as monoidal categories. Moreover if is a finite-dimensional Hopf algebra with bijective antipode, the isomorphism is braided. Finally, as an application of this category isomorphism, we give two results.
In this paper, we study the type Hopf algebras and present its braided and quasitriangular Hopf algebra structure. This generalizes well-known results on and type Hopf algebras. Finally, the classification of type Hopf algebras is given.
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