Journal of Algebraic Combinatorics最新文献

Given integers (k,lge 2), where either l is odd or k is even, let n(k, l) denote the largest integer n such that each element of (A_n) is a product of k many l-cycles. M. Herzog, G. Kaplan and A. Lev conjectured that (lfloor frac{2kl}{3} rfloor le n(k,l)le lfloor frac{2kl}{3}rfloor +1) [Herzog et al. in J Combin Theory Ser A, 115:1235-1245 2008]. It is known that the conjecture holds when (k=2,3,4). Moreover, it is also true when (3mid l). In this article, we determine the exact value of n(k, l) when (3not mid l) and (kge 5). As an immediate consequence, we get that (n(k,l)<lfloor frac{2kl}{3}rfloor ) when (kge 5) and (3not mid l), which shows that the above conjecture is not true in general. In fact in this case, the difference between the exact value of n(k, l) and the conjectured value grows linearly in terms of k. Our results complete the determination of n(k, l) for all values of k and l.

Abstract

We introduce the notion of an anti-dendriform algebra as a new approach of splitting the associativity. It is characterized as the algebra with two multiplications giving their left and right multiplication operators, respectively, such that the opposites of these operators define a bimodule structure on the sum of these two multiplications, which is associative. This justifies the terminology due to a closely analogous characterization of a dendriform algebra. The notions of anti- ({mathcal {O}}) -operators and anti-Rota–Baxter operators on associative algebras are introduced to interpret anti-dendriform algebras. In particular, there are compatible anti-dendriform algebra structures on associative algebras with nondegenerate commutative Connes cocycles. There is an important observation that there are correspondences between certain subclasses of dendriform and anti-dendriform algebras in terms of q-algebras. As a direct consequence, we give the notion of Novikov-type dendriform algebras as an analogue of Novikov algebras for dendriform algebras, whose relationship with Novikov algebras is consistent with the one between dendriform and pre-Lie algebras. Finally, we extend to provide a general framework of introducing the notions of analogues of anti-dendriform algebras, which interprets a new splitting of operations.

We consider networks on minimal Cayley graphs of irreducible complex reflection groups G(m, p, n). We show that resistance diameters of these graphs have asymptotic (Theta (1/n)) as (nrightarrow infty ) under fixed m, p. Non-trivial lower and upper asymptotic bounds for critical probabilities of percolation for there appearing a giant connected component have been obtained.

Let ({mathcal {A}}subseteq {[n]atopwithdelims ()a}) and ({mathcal {B}}subseteq {[n]atopwithdelims ()b}) be two families of subsets of [n], we say ({mathcal {A}}) and ({mathcal {B}}) are cross-intersecting if (Acap Bne emptyset ) for all (Ain {mathcal {A}}), (Bin {mathcal {B}}). In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.

Abstract

Given a finite abelian group G and cyclic subgroups A, B, C of G of the same order, we find necessary and sufficient conditions for A, B, C to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups, we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.

A mixed dihedral group is a group H with two disjoint subgroups X and Y, each elementary abelian of order (2^n), such that H is generated by (Xcup Y), and (H/H'cong Xtimes Y). In this paper, we give a sufficient condition such that the automorphism group of the Cayley graph (textrm{Cay}(H,(Xcup Y){setminus }{1})) is equal to (Hrtimes A(H,X,Y)), where A(H, X, Y) is the setwise stabiliser in ({{,textrm{Aut},}}(H)) of (Xcup Y). We use this criterion to resolve a question of Li et al. (J Aust Math Soc 86:111-122, 2009), by constructing a 2-arc-transitive normal cover of order (2^{53}) of the complete bipartite graph ({{textbf {K}}}_{16,16}) and prove that it is not a Cayley graph.

Let (D_{n,k}) be the set of all permutations of the symmetric group (S_n) that have no cycles of length i for all (1 le i le k). In the paper mentioned above, Ku, Lau, and Wong prove that the set of all the largest independent sets of the Cayley graph (text {Cay}(S_n,D_{n,k})) is equal to the set of all the largest independent sets in the derangement graph (text {Cay}(S_n,D_{n,1})), provided n is sufficiently large in terms of k. We give a simpler proof that holds for all n, k and also applies to the alternating group.

In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.

For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for q-matroids, the q-analogue of matroids. This is a lot less straightforward than in the classical case, as we will try to convince the reader. With the use of submodular functions and the q-analogue of matroid union we come to a definition of the direct sum of q-matroids. As a motivation for this definition, we show it has some desirable properties.

We give a rank augmentation technique for rank three string C-group representations of the symmetric group (S_n) and list the hypotheses under which it yields a valid string C-group representation of rank four thereof.