Annals of Combinatorics最新文献

Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set S of vertices of an n-vertex graph G such that (G - N[S]), the graph obtained by deleting the closed neighborhood of S, contains no vertices. In the proof of the Art Gallery Theorem, Chvátal showed that the minimum size, called the domination number of G and denoted by (gamma (G)), is at most n/3 if G is a mop. Here we consider a modification by allowing (G - N[S]) to have a maximum degree of at most k. Let (iota _k(G)) denote the size of a smallest set S for which this is achieved. If (n le 2k+3), then trivially (iota _k(G) le 1). Let G be a mop on (n ge max {5,2k+3}) vertices, (n_2) of which are of degree 2. Upper bounds on (iota _k(G)) have been obtained for (k = 0) and (k = 1), namely (iota _{0}(G) le min {frac{n}{4},frac{n+n_2}{5},frac{n-n_2}{3}}) and (iota _1(G) le min {frac{n}{5},frac{n+n_2}{6},frac{n-n_2}{3}}). We prove that (iota _{k}(G) le min {frac{n}{k+4},frac{n+n_2}{k+5},frac{n-n_2}{k+2}}) for any (k ge 0). For the original setting of the Art Gallery Theorem, the argument presented yields that if an art gallery has exactly n corners and at least one of every (k + 2) consecutive corners must be visible to at least one guard, then the number of guards needed is at most (n/(k+4)). We also prove that (gamma (G) le frac{n - n_2}{2}) unless (n = 2n_2), (n_2) is odd, and (gamma (G) = frac{n - n_2 + 1}{2}). Together with the inequality (gamma (G) le frac{n+n_2}{4}), obtained by Campos and Wakabayashi and independently by Tokunaga, this improves Chvátal’s bound. The bounds are sharp.

A weak composition of an integer s with m parts is a way of writing s as the sum of a sequence of non-negative integers of length m. Given two positive integers m and n, let N(m, n) denote the set of all weak compositions (alpha =(alpha _1,dots ,alpha _m)) with (0 le alpha _i le n) for (1 le i le m) and (c_w^{m,n}(s)) be the number of weak composition of s into m parts with no part exceeding n. A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. In this paper, we show that the poset N(m, n) can be expressed as a disjoint of symmetric chains by constructive method, which implies that its rank sequence (c_w^{m,n}(0),c_w^{m,n}(1),dots ,c_w^{m,n}(mn)) is unimodal and symmetric.

In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set (P_r) of r-element multichains of P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there are exactly (2^r) such functions which yield subdivisions of the order complex of P, of which (2^{r-1}) are pairwise different. Within this class are, for example, the order complexes of the intervals in P, the zig-zag poset of P, and the (r{hbox {th}}) edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are order complexes and homotopy equivalent to the order complex of P.

We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.

Motivated by a question of Defant and Propp (Electron J Combin 27:Article P3.51, 2020) regarding the connection between the degrees of noninvertibility of functions and those of their iterates, we address the combinatorial optimization problem of minimizing the sum of squares over partitions of n with a nonnegative rank. Denoting the sequence of the minima by ((m_n)_{nin {mathbb {N}}}), we prove that (m_n=Theta left( n^{4/3}right) ). Consequently, we improve by a factor of 2 the lower bound provided by Defant and Propp for iterates of order two.

Let ({{mathcal {T}}}_{d}(n)) be the set of d-ary rooted trees with n internal nodes. We give a method to construct a sequence (( textbf{t}_{n},nge 0)), where, for any (nge 1), ( textbf{t}_{n}) has the uniform distribution in ({{mathcal {T}}}_{d}(n)), and ( textbf{t}_{n}) is constructed from ( textbf{t}_{n-1}) by the addition of a new node, and a rearrangement of the structure of ( textbf{t}_{n-1}). This method is inspired by Rémy’s algorithm which does this job in the binary case, but it is different from it. This provides a method for the random generation of a uniform d-ary tree in ({{mathcal {T}}}_{d}(n)) with a cost linear in n.

This paper investigates partitions which have neither parts nor hook lengths divisible by (p), referred to as (p)-core (p')-partitions. We show that the largest (p)-core (p')-partition corresponds to the longest walk on a graph with vertices ({0, 1, ldots , p-1}) and labelled edges defined via addition modulo (p). We also exhibit an explicit family of large (p)-core (p')-partitions, giving a lower bound on the size of the largest such partition which is of the same degree as the upper bound found by McSpirit and Ono.

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the kth moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the (p^m)th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak.

In this article, we undertake the problem of finding the first four trees on a fixed number of vertices with the maximum smallest positive eigenvalue. Let ({mathcal {T}}_{n,d}) denote the class of trees on n vertices with diameter d. First, we obtain the bounds on the smallest positive eigenvalue of trees in ({mathcal {T}}_{n,d}) for (d =2,3,4) and then upper bounds on the smallest positive eigenvalue of trees are obtained in general class of all trees on n vertices. Finally, the first four trees on n vertices with the maximum, second maximum, third maximum and fourth maximum smallest positive eigenvalue are characterized.

In this paper, first we introduce the notion of two-sided Cayley graph of a semigroup. Then, we investigate some fundamental properties of these graphs and we use our results to give partial answers to some problems raised by Iradmusa and Praeger about two-sided group graphs (two-sided Cayley graphs of groups). Specially, as a consequence of our results, we determine all undirected two-sided Cayley graphs of groups which are connected. Furthermore, by introducing the notion of color-preserving automorphisms of a two-sided Cayley graph of a semigroup (group) and calculating them under some assumptions, we determine the family of color-vertex transitive two-sided Cayley graphs of semigroups (groups).