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《Discrete Mathematics》2022,345(8):112904
Let be the minimum integer such that every plane graph with girth g at least , minimum degree and no -paths consisting of vertices of degree 2, where , has a 3-vertex with at least t neighbors of degree 2, where .In 2015, Jendrol' and Maceková proved . Later on, Hudák et al. established , Jendrol', Maceková, Montassier, and Soták proved , and , and we recently proved that and .Thus is already known for and all t. In this paper, we prove that , , and whenever . 相似文献
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《Discrete Mathematics》2022,345(11):113023
Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an -regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in has exactly b neighbors in C. In particular, -regular sets and -regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an -regular set of G if there exists a Cayley graph of G which admits C as an -regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a -regular set of G, then it must also be an -regular set of G for any and such that a is even when is odd. A similar result involving -regular sets of such groups is also obtained in the paper. 相似文献
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In this paper we completely classify the linearly full homogeneous holomorphic two-spheres in the complex Grassmann manifolds and . We also obtain the Gauss equation for the holomorphic immersions from a Riemann surface into . By using which, we give explicit expressions of the Gaussian curvature and the square of the length of the second fundamental form of these homogeneous holomorphic two-spheres in and . 相似文献
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《Discrete Mathematics》2021,344(12):112601
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Lawler, Schramm, and Werner gave in 2003 an explicit formula of the probability that does not intersect a deterministic hull. For general with , no such explicit formula has been obtained so far. In this paper, we shall consider a random hull generated by an independent chordal conformal restriction measure and obtain an explicit formula for the probability that does not intersect this random hull for any . As a corollary, we will give a new proof of Werner's result on conformal restriction measures. 相似文献
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Yong Liu 《Annals of Pure and Applied Logic》2019,170(4):515-538
There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called -cupping degree and show that there exists an isolated -cupping degree, and there exists a proper -cupping degree. It helps understanding various degree structures in the Ershov Hierarchy. 相似文献