The billiard "table" is modeled as an n-dimensional box [0, a(1)] & times; [0, a(2)] & times; & centerdot; & centerdot; & centerdot; & times; [0, a(n)] subset of R-n, with each side having real-valued lengths a(i) that are pairwise commensurable. A ball is launched from the origin in direction d = (1, 1, ... , 1). The ball is reflected if it hits the boundary of the box. It comes to a halt when reaching a corner. We show that the number of intersections of the billiard curve at any given point in the box is either 0 or a power of 2. To prove this, we use algebraic and number theoretic tools to establish a bijection between the set of intersections at a given point of the billiard curve and the set of satisfying assignments of a specific constraint satisfaction problem.
In graph theory, the center function identifies a set of vertices in a connected graph that minimizes the maximum distance from any other vertex. Through an axiomatic characterization of the center function, we identify specific axioms that characterize its behaviour on connected graphs. Universal axioms are those axioms satisfied by the center function on all connected graphs. However, on certain graphs, these universal axioms are insufficient to completely characterize the center function. Non-universal axioms, specific to particular graph classes, were introduced to address this limitation. This work focuses on finding an axiomatic characterization of the center function on graphs with diameter two, employing a combination of both universal and non-universal axioms.
In this paper, motivated by the definition of a non-commutative harmonic oscillator studied by Parmeggiani and Wakayama [Proc. Natl. Acad. Sci. USA 98 (2001), 26-30], we introduce some variants of the Euler operator with the aim of exploring the similarity between recurrence relations and differential operators. By using variants of the Euler operator, we establish a link between Eulerian polynomials of types A and B, and second-order Eulerian polynomials of types A and B as well as Narayana polynomials of types A and B.
Let G ' be a subgraph of a simple finite graph G and U be a subset of a set V. We say that a lambda-fold G '-design (U, C) of order u is embedded into a & micro;- fold G-design (V, B) of order u+w, & micro; >= lambda, if there is an injective function f : C -> B such that C is a subgraph of f (C) for every C is an element of C. The mapping f is called the embedding of (U, C) into (V, B). If w attains the minimum possible value, then f is a minimum embedding. In this paper a complete solution is given to the problem of determining a minimum embedding of a lambda-fold P3-design into a lambda-fold kite system (lambda = & micro;) for any index lambda.
A dominator coloring (DC) of a graph G is a proper coloring in which every vertex dominates all vertices of at least one color class. Let chi(d)(G) denote the minimum number of colors in a DC of G. Arumugam et al. in 2012 proved that chi(d)(G) + 1 = 8. In this paper, by determining the exact values of chi(d)(& micro;(C8)) and chi(d)(& micro;(C9)), we provide counterexamples to the above two results. Motivated by this, we completely characterize when chi(d)(& micro;(C-n)) = chi(d)(C-n) + 2, namely, if and only if n Xi 3 (mod 6) with n not equal 3, or n Xi 5 (mod 6). In addition, we determine the exact values of chi(d)(& micro;(P-n)) for all paths P-n. Our results correct earlier claims and provide a complete dethese graph classes.
We study two types of generalizations of the Narayana array. Type 1 counts the number of Dyck paths of semilength n with k peaks (or valleys) whose heights belong to a given set S, whereas Type 2 counts Dyck paths of semilength n with exactly k peaks (or valleys), all of which have heights in S. We focus on the following sets: S = {m}, S = {m, m + 1}, the non-negative even heights, the positive odd heights, and more generally heights congruent to a fixed residue modulo a. For each case, we provide a bivariate generating function f (x, y) that characterizes the corresponding Narayana array, establish combinatorial connections through bijections with other combinatorial structures, and obtain asymptotic approximations for the expected number of peaks or valleys of height in S in Dyck paths of a given semilength.
The even-linked double star S-2p(n, m), where n >= m >= 0, is the graph formed by the union of two stars K-1,K-n and K-1,K-m connected by a path of 2p vertices between their centers. Its Ramsey number r(S-2p(n, m)) is the smallest integer r such that every 2-edge-coloring of the edges of a K-r admits a monochromatic S-2p(n, m). In this paper, we establish bounds on the value of r(S-2p(n, m)). In particular, we obtain r(S-2p(n, m)) <= 2(n + m + p-1) for n >= 2p >= m and determine the exact value of r(S-2p(n, 2)) when p >= n.
We study generalized recursive trees, that is, recursive trees where the nodes could recruit batches of children. We assume the size of the batch at each step follows the distribution of a generic random variable X is an element of {x(1), x(2), x(3), ...} subset of N, with P(X = x(j)) = p(j) > 0, for j = 1, 2, 3, .... In this notation x(1) < x(2) < x(3) < & centerdot;& centerdot;& centerdot;. The support of X imposes certain admissible node outdegrees. We look at the joint distribution of the first three admissible outdegrees (the zero is included because a tree always has leaves). The values of x(1) and x(2) fall in three regimes in the first quadrant of the two-dimensional space. Namely, the regimes are 2x(1) < x(2), 2x(1) = x(2), and 2x(1) > x(2). In all three regimes, we have asymptotic trivariate normal distributions for the counts of the first three outdegrees, with marked differences in the asymptotics of the vector of means and the covariance matrix. Polya urns with random replacement matrices are our chief tool. We conclude the paper with an example from the middle regime, where the additions follow a distribution as one plus a Poisson random variable with parameter 1.
Let G be a bipartite graph with (X,Y) as its bipartition. We say that G(X,Y) has a k packing in a complete bipartite graph B-n(V-1,V-2) of order n with X subset of V(1)and Y subset of V-2 if there exists a sequence of permutations of V(G), say Phi = (phi 1,...,phi k), such that phi(i)(X) subset of V-1, phi(i)(Y) subset of V-2 and E(phi i(G)) boolean AND E(phi(j)(G)) = & empty; for all i,j is an element of {1,...,k} and i= j. In this paper, we say that a 2packing of G(X,Y) in B-n(V-1,V-2 ) is a bipacking of G(X,Y) in B-n(V-1,V-2 ). We will show that if G is a bipartite graph of order n with girth at least 8, then there is a complete bipartite graph Bn+1 such that there is a bipacking of G in Bn+1. The condition on the girth of G is sharp. This proves the conjecture proposed by Wang in 2019. Furthermore, we conjecture that for a bipartite graph G with order n and girth at least 4k where k >= 2, there is a k packing of G in some Bn+k-1.
In this paper, we determine all r-regular graphs whose second largest eigenvalue lambda(2) satisfies lambda(2) <= 1, under the condition n >= 2r, where n denotes the number of vertices. As an application, we provide a complete classification of all r-regular graphs with lambda(2) <= 1 for 9 <= r <= 10. This result extends the previous classifications given by Stanic [ Lin. Multilin. Alg. 58 (2010), 545-554] and by Koledin and Stanic, [ Novi Sad J. Math. 43 (2013), 145-153], which fully resolved the cases r <= 8.
Let G be a simple graph. The diagonal graph Ramsey number R(G, G) is defined to be the minimum n, where every 2-coloring of the edges of K-n contains a red G or a blue G. In this paper, new diagonal graph Ramsey numbers are calculated for some classes of even cycles with pendant edges.
A Grand Dyck path of semilength n with m flaws is a path in the integer lattice which starts at the origin and consists of n up steps U = (1, 1) and n down steps D = (1, -1), and that has exactly m up steps below the line y = 0. The classical Chung-Feller theorem asserts that the number of grand Dyck paths of semilength n with m flaws is the nth Catalan number and is independent of m. In this paper, by using a bijection and generating functions, we prove a refinement of the Chung-Feller theorem: the number of Grand Dyck paths of semilength n having m flaws and k descents is the Narayana number N-n,N-k, and is independent of m. We also enumerate the Grand Dyck paths ending with a down step or an up step, and obtain some interesting results related to the Narayana numbers or Catalan numbers.
A d-dimensional nowhere-zero r-flow on a graph G, an (r, d)-NZF from now on, is a flow where the value on each edge is an element of R-d whose (Euclidean) norm lies in the interval [1, r-1]. Such a notion is a natural generalization of the well-known concept of a circular nowhere-zero r-flow (i.e., d = 1). The minimum of the real numbers r such that a graph G admits an (r, d)-NZF is called the d-dimensional flow number of G and is denoted by Phi(d)(G). In this paper we provide a geometric description of some d-dimensional flows on a graph G, and we prove that the existence of a suitable cycle double cover of G is equivalent, for G, to admit such a geometrically constructed (r, d)-NZF. This geometric approach allows us to provide upper bounds for Phi(d-2)(G) and Phi(d-1)(G), assuming that G admits an (oriented) d-cycle double cover.
For each t >= 1 let W-t denote the class of graphs other than stars that have diameter 2 and contain neither a triangle nor a K-2,K-t. The famous Hoffman-Singleton Theorem implies that W-2 is finite. Recently Wood suggested the study of W-t for t > 2 and conjectured that W-t is finite for all t >= 2. In this note we show that (1) W-3 is infinite, (2) W-5 contains infinitely many regular graphs, and (3) W-7 contains infinitely many Cayley graphs. Our W-3 and W-5 examples are based on so-called crooked graphs, first constructed by de Caen, Mathon, and Moorhouse. Our W-7 examples are Cayley graphs with vertex set F-p(2) for prime p equivalent to 11 (mod 12). We also highlight the surprising fact that crooked graphs themselves provide an infinite family of graphs which imply that ex(n, {C-3, K-2,K-3}) = (1/root 2 + o(1))n(3/2) for an infinite, albeit sparse, set of n's.
In recent work, Amdeberhan and Merca considered the integer partition function a(n) which counts the number of integer partitions of weight n wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of three colors. One of the results that they proved was that, for all n >= 0, a(7n + 2) equivalent to 0 (mod 7). In this work, we generalize this function a(n) by naturally placing it within an infinite family of related partition functions. Using elementary generating function manipulations and classical q-series identities, we then prove infinitely many congruences modulo 7 which are satisfied by members of this family of functions.
Given a direct sum A of full matrix algebras, if there is a combinatorial interpretation associated with both the dimension of A and the dimensions of the irreducible A-modules, then this can be thought of as providing an analogue of the famous Frobenius-Young identity n! = Sigma(lambda proves n )(f( lambda))(2) derived from the semisimple structure of the symmetric group algebra CSn, letting f( lambda) denote the number of standard Young tableaux of partition shape lambda proves n. By letting g(alpha) denote the number of standard immaculate tableaux of composition shape alpha (sic) n, we construct an algebra CJ(n) with a semisimple structure such that dim CJ(n) = Sigma(alpha(sic)n )(g(alpha))(2) and such that CJ(n) contains an isomorphic copy of CSn. We bijectively prove a recurrence for dim CJ(n) so as to construct a basis of CJ(n )indexed by permutation-like objects that we refer to as immacutations. We form a basis B-n of CJ(n )such that CBn has the structure of a monoid algebra in such a way so that B-n is closed under the multiplicative operation of CJ(n), yielding a monoid structure on the set of order-n immacutations.
A perfect 1-factorisation of a graph is a decomposition of that graph into 1-factors such that the union of any two 1-factors is a Hamiltonian cycle. A Latin square of order n is row-Hamiltonian if for every pair (r,s) of distinct rows, the permutation mapping r to s has a single cycle of length n. We report the results of a computer enumeration of the perfect 1-factorisations of the complete bipartite graph K_11,11. This also allows us to find all row-Hamiltonian Latin squares of order 11. Finally, we plug a gap in the literature regarding how many row-Hamiltonian Latin squares are associated with the classical families of perfect 1-factorisations of complete graphs.
The design spectrum of a simple graph G is the set of positive integers n such that there exists an edgewise decomposition of the complete graph K-n into n(n-1)/(2|E(G)|) copies of G. The purpose of this short paper is to prove that the Shrikhande graph and the line graph of K-4,K-4 have the design spectrum {96t + 1: t = 0,1, 2, ... }.
The anti-van der Waerden number of a graph G is the fewest number of colors needed to guarantee a rainbow 3-term arithmetic progression in any vertex coloring of G, denoted aw(G, 3). It is known that the anti-van der Waerden number of graph products is 3 <= aw(G square H, 3) <= 4. Previous work has been done on classifying families of graph products into aw(G square H, 3) = 3 and aw(G square H, 3) = 4. Some of these families include the product of two paths, the product of paths and cycles, the product of two cycles, and the product of odd cycles with any graph. Recently, a partial characterization of the product of two trees was established. This paper completes the characterization for aw(T square T ', 3) where T and T' are trees. Moreover, this result extends to a full classification of products of forests.
Analogously to de Bruijn sequences, Orientable sequences have application in automatic position-location applications and, until recently, studies of these sequences focused on the binary case. In recent work by Alhakim et al., recursive methods of construction were described for orientable sequences over arbitrary finite alphabets, requiring 'starter sequences' with special properties. Some of these methods required as input special orientable sequences, i.e. orientable sequences which were simultaneously negative orientable. We exhibit methods for constructing special orientable sequences with properties appropriate for use in two of the recursive methods of Alhakim et al. As a result we are able to show how to construct special orientable sequences for arbitrary sizes of alphabet (larger than a small lower bound) and for all window sizes. These sequences have periods asymptotic to the optimal as the alphabet size increases.
