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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》2023,346(5):113344
For any positive integer k, let denote the least integer such that any n-vertex graph has an induced subgraph with at least vertices, in which at least vertices are of the same degree. Caro, Shapira and Yuster initially studied this parameter and showed that . For the first nontrivial case, the authors proved that , and the exact value was left as an open problem. In this paper, we first show that , improving the former result as well as a recent result of Kogan. For special families of graphs, we prove that for -free graphs, and for large -free graphs. In addition, extending a result of Erd?s, Fajtlowicz and Staton, we assert that every -free graph is an induced subgraph of a -free graph in which no degree occurs more than three times. 相似文献
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We study the stationary Stokes system in divergence form. The coefficients are assumed to be merely measurable in one direction and have Dini mean oscillations in the other directions. We prove that if is a weak solution of the system, then is bounded and its certain linear combinations are continuous. We also prove a weak type- estimate for under a stronger assumption on the -mean oscillation of the coefficients. The corresponding results up to the boundary on a half ball are also established. These results are new even for elliptic equations and systems. 相似文献
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We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form in which x and τ represent dimensionless distance and time respectively and is a parameter related to the relaxation time. Furthermore the reaction function, , is given by the bistable cubic polynomial, in which is a parameter. The initial data is given by a simple step function with for and for . It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front which is either of reaction–diffusion or of reaction–relaxation type. The one exception to this occurs when in which case the large time attractor for the solution of the initial-value problem is a stationary state solution of kink type centred at the origin. 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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We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below , and not above . This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if , , and are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each . The resulting structure is an example of a structure that has a degree of categoricity, but not strongly. 相似文献
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《Discrete Mathematics》2021,344(12):112601
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《Journal of Pure and Applied Algebra》2019,223(11):5030-5048
Take positive integers m, n and d. Let Y be an m-fold cyclic cover of ramified over a general hypersurface of degree md. In this paper we study the space of lines in Y and show that it is smooth of dimension if and . When , our result gives a formula on the number of m-contact order lines of X (see Definition 1.2). 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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Relatively recently it was proved that if Γ is an arbitrary set, then any equivalent norm on can be approximated uniformly on bounded sets by polyhedral norms and smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the ‘discrete’ Lorentz spaces , and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number α, there exists a scattered compact space K having Cantor–Bendixson height at least α, such that every equivalent norm on can be approximated as above. 相似文献