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《Journal of Pure and Applied Algebra》2023,227(2):107190
Let W be a finite Coxeter group and X a subset of W. The length polynomial is defined by , where ? is the length function on W. If then we call the involution length polynomial of W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W. In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type and . Moreover, we give a counterexample to a unimodality conjecture stated in [11]. 相似文献
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This paper deals with the chemotaxis-growth system: , , in a smooth bounded domain with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution exponentially stabilizes to the constant stationary solution in the norm of as provided that and any given nonnegative and suitably smooth initial data fulfills , which extends the condition in [8]. 相似文献
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《Discrete Mathematics》2021,344(12):112604
A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index of G is Δ or . A graph G is class 1 if , and class 2 if ; G is Δ-critical if it is connected, class 2 and for every . A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, for every . Such graphs have intimate relation to -co-critical graphs, where a non-complete graph G is -co-critical if there exists a k-coloring of such that G does not contain a monochromatic copy of but every k-coloring of contains a monochromatic copy of for every . We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all -co-critical graphs. We prove that if G is a -co-critical graph on vertices, then where ε is the remainder of when divided by 2. This bound is best possible for all and . 相似文献
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《Discrete Mathematics》2022,345(9):112945
The coinvariant algebra is a quotient of the polynomial ring whose algebraic properties are governed by the combinatorics of permutations of length n. A word over the positive integers is packed if whenever appears as a letter of w, so does . We introduce a quotient of which is governed by the combinatorics of packed words. We relate our quotient to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring. 相似文献
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《Discrete Mathematics》2022,345(8):112903
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number is the least integer k for which G admits a coloring with k colors such that each color class induces a -degenerate subgraph of G. So is the chromatic number and is the point arboricity. The point partition number with was introduced by Lick and White. A graph G is called -critical if every proper subgraph H of G satisfies . In this paper we prove that if G is a -critical graph whose order satisfies , then G can be obtained from two non-empty disjoint subgraphs and by adding t edges between any pair of vertices with and . Based on this result we establish the minimum number of edges possible in a -critical graph G of order n and with , provided that and t is even. For the corresponding two results were obtained in 1963 by Tibor Gallai. 相似文献
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《Discrete Mathematics》2022,345(3):112717
A transversal set of a graph G is a set of vertices incident to all edges of G. The transversal number of G, denoted by , is the minimum cardinality of a transversal set of G. A simple graph G with no isolated vertex is called τ-critical if for every edge . For any τ-critical graph G with , it has been shown that by Erd?s and Gallai and that by Erd?s, Hajnal and Moon. Most recently, it was extended by Gyárfás and Lehel to . In this paper, we prove stronger results via spectrum. Let G be a τ-critical graph with and , and let denote the largest eigenvalue of the adjacency matrix of G. We show that with equality if and only if G is , , or , where ; and in particular, with equality if and only if G is . We then apply it to show that for any nonnegative integer r, we have and characterize all extremal graphs. This implies a pure combinatorial result that , which is stronger than Erd?s-Hajnal-Moon Theorem and Gyárfás-Lehel Theorem. We also have some other generalizations. 相似文献
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