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《Discrete Mathematics》2022,345(8):112902
For a simple graph G, denote by n, , and its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if and for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that must be edge-chromatic critical if , and they verified this when . In this paper, we prove it for . 相似文献
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Shai Shechter 《Journal of Pure and Applied Algebra》2019,223(10):4384-4425
Let be a complete discrete valuation ring with finite residue field of odd characteristic, and let G be a symplectic or special orthogonal group scheme over . For any let denote the ?-th principal congruence subgroup of . An irreducible character of the group is said to be regular if it is trivial on a subgroup for some ?, and if its restriction to consists of characters of minimal -stabilizer dimension. In the present paper we consider the regular characters of such classical groups over , and construct and enumerate all regular characters of , when the characteristic of is greater than two. As a result, we compute the regular part of their representation zeta function. 相似文献
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A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. Let be the finite field with q elements where q is a power of an odd prime number. The quadratic forms graph, denoted by where , has all quadratic forms on as vertices and two vertices f and g are adjacent whenever or 2. We prove that every is a pseudo-core. Further, when n is even, is a core. When n is odd, is not a core. On the other hand, we completely determine the independence number of . 相似文献
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《Discrete Mathematics》2022,345(8):112917
Let and denote the flow number and the circular flow number of a flow-admissible signed graph , respectively. It is known that for every unsigned graph G. Based on this fact, in 2011 Raspaud and Zhu conjectured that holds also for every flow-admissible signed graph . This conjecture was disproved by Schubert and Steffen using graphs with bridges and vertices of large degree. In this paper we focus on cubic graphs, since they play a crucial role in many open problems in graph theory. For cubic graphs we show that if and only if and if , then . We also prove that all pairs of flow number and circular flow number that fulfil these conditions can be achieved in the family of bridgeless cubic graphs and thereby disprove the conjecture of Raspaud and Zhu even for bridgeless signed cubic graphs. Finally, we prove that all currently known flow-admissible graphs without nowhere-zero 5-flow have flow number and circular flow number 6 and propose several conjectures in this area. 相似文献
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Let be a Noetherian local ring and M a finitely generated R-module. The invariants and of M were introduced in [3] and [17] in order to measure the non-Cohen–Macaulayness and the non-sequential-Cohen–Macaulayness of M, respectively. Let be the filtration of M such that is the largest submodule of M of dimension less than for all and . In this paper we prove that if , then there exists a constant c such that for all good parameter ideals of M with respect to this filtration. Here is the reducibility index of on M. This is an extension of the main results of [19], [20], [24]. 相似文献
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《Journal of Pure and Applied Algebra》2023,227(3):107228
Let be a finite permutation group and recall that the base size of G is the minimal size of a subset of Ω with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form with soluble point stabilisers, where , and is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form and we classify the groups with a unique regular suborbit under the assumption that P is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups. 相似文献
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《Discrete Mathematics》2022,345(7):112893
In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let Γ and be finite simple graphs with at least three vertices such that there exists a bijective map and for any , there exists an isomorphism . Then we define the associated directed graph with two kinds of arrows from the graphs Γ and , the bijective map f and the isomorphisms . By investigating the associated directed graph , we study when are the two graphs Γ and isomorphic. 相似文献
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《Journal of Pure and Applied Algebra》2022,226(10):107074
For a commutative ring A we consider a related graph, , whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that is path-connected if and only if A is a -ring, in the terminology of P. M. Cohn. Furthermore, if denotes the clique complex of , we prove that is simply connected if and only if A is universal for . More precisely, our main theorem is that for any commutative ring A the fundamental group of is isomorphic to the group modulo the subgroup generated by symbols. 相似文献
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