An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS(v)”. Our main idea is to use transitive resolvable idempotent symmetric quasigroups instead of TKTS. As an application, we can combine the known results on LKTS and LR-designs to obtain the existence of an LKTS(3ⁿm(2·13^n₁+1)(2·13^n_t+1)) for n1, m{1,5,11,17,25,35,43,67,91,123}{2^2r+125^s+1 : r0,s0}, t0 and n_i1 (i=1,…,t).     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

On L(d,1)-labelings of graphs

Given a graph G and a positive integer d, an L(d,1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u)−f(v)|d; if u and v are not adjacent but there is a two-edge path between them, then |f(u)−f(v)|1. The L(d,1)-number of G, λ_d(G), is defined as the minimum m such that there is an L(d,1)-labeling f of G with f(V){0,1,2,…,m}. Motivated by the channel assignment problem introduced by Hale (Proc. IEEE 68 (1980) 1497–1514), the L(2,1)-labeling and the L(1,1)-labeling (as d=2 and 1, respectively) have been studied extensively in the past decade. This article extends the study to all positive integers d. We prove that λ_d(G)Δ²+(d−1)Δ for any graph G with maximum degree Δ. Different lower and upper bounds of λ_d(G) for some families of graphs including trees and chordal graphs are presented. In particular, we show that the lower and the upper bounds for trees are both attainable, and the upper bound for chordal graphs can be improved for several subclasses of chordal graphs.     

Further results on nearly Kirkman triple systems with subsystems

In this paper we further discuss the embedding problem for nearly Kirkman triple systems and get the result that: (1) For u≡v≡0 (mod 6), v78, and u3.5v, there exists an NKTS(u) containing a sub-NKTS(v). (2) For v=18,24,30,36,42,48,54,60,66 or 72, there exists an NKTS(u) containing a sub-NKTS(v) if and only if u≡0 (mod 6) and u3v.     

A doubly cyclic channel assignment problem

A standard model for radio channel assignment involves a set V of sites, the set {0,1,2,…} of channels, and a constraint matrix (w(u, v)) specifying minimum channel separations. An assignment f:V→{0,1,2,…} is feasible if the distance f(u) − f(v)w(u, v) for each pair of sites u and v. The aim is to find the least k such that there is a feasible assignment using only the k channels 0, 1, …, k − 1, and to find a corresponding optimal assignment.

We consider here a related problem involving also two cycles. There is a given cyclic order τ on the sites, and feasible assignments f must also satisfy f(τv) f(v) for all except one site v. Further, the channels are taken to be evenly spaced around a circle, so that if the k channels 0, 1, …, k − 1 are available then the distance between channels i and j is the minimum of ¦i − j¦ and k − ¦i − j¦. We show how to find a corresponding optimal channel assignment in O(¦V¦³) steps.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

New families of exact solitary patterns solutions for the nonlinearly dispersive R(m,n) equations

The R(m,n) equations u_tt+a(uⁿ)_xx+b(u^m)_xxtt=0 (a, b const) are investigated by using some ansatze. As a result, new exact solitary patterns solutions and solitary wave solutions are obtained. These obtained solutions show that not only the nonlinearly dispersive R(m,m) equations (m≠1) but also the linearly dispersive R(1,n) equations (m=1) possess solitary patterns solutions, which has infinite slopes or cusps and solitary wave solutions.     

Maximum number of edges in a critically k-connected graph

A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any vV(G). We show that if n,k are integers with k4 and nk+2, and G is a critically k-connected graph of order n, then |E(G)|n(n−1)/2−p(n−k)+p²/2, where p=(n/k)+1 if n/k is an odd integer and p=n/k otherwise. We also characterize extremal graphs.     

Some results on integral sum graphs

Let Z denote the set of all integers. The integral sum graph of a finite subset S of Z is the graph (S,E) with vertex set S and edge set E such that for u,vS, uvE if and only if u+vS. A graph G is called an integral sum graph if it is isomorphic to the integral sum graph of some finite subset S of Z. The integral sum number of a given graph G, denoted by ζ(G), is the smallest number of isolated vertices which when added to G result in an integral sum graph. Let x denote the least integer not less than the real x. In this paper, we (i) determine the value of ζ(K_n−E(K_r)) for r2n/3−1, (ii) obtain a lower bound for ζ(K_n−E(K_r)) when 2r<2n/3−1 and n5, showing by construction that the bound is sharp when r=2, and (iii) determine the value of ζ(K_r,r) for r2. These results provide partial solutions to two problems posed by Harary (Discrete Math. 124 (1994) 101–108). Finally, we furnish a counterexample to a result on the sum number of K_r,s given by Hartsfiedl and Smyth (Graphs and Matrices, R. Rees (Ed.), Marcel, Dekker, New York, 1992, pp. 205–211).     

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type

(|u′|^m−2u′)′ + f(t,u,u′)=0, m 2

are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.     

Algorithmic complexity of list colorings

Given a graph G = (V,E) and a finite set L(v) at each vertex v ε V, the List Coloring problem asks whether there exists a function f:V → _vεVL(V) such that (i) f(v)εL(v) for each vεV and (ii) f(u) ≠f(v) whenever u, vεV and uvεE. One of our results states that this decision problem remains NP-complete even if all of the followingconditions are met: (1) each set L(v) has at most three elements, (2) each “color” xε_vεVL(v) occurs in at most three sets L(v), (3) each vertex vεV has degree at most three, and (4) G is a planar graph. On the other hand, strengthening any of the assumptions (1)–(3) yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.     

An extension theorem for t-designs

In 1994, van Trung (Discrete Math. 128 (1994) 337–348) [9] proved that if, for some positive integers d and h, there exists an S_λ(t,k,v) such that

then there exists an S_λ(v−t+1)(t,k,v+1) having v+1 pairwise disjoint subdesigns S_λ(t,k,v). Moreover, if B_i and B_j are any two blocks belonging to two distinct such subdesigns, then d|B_i∩B_j|<k−h. In 1999, Baudelet and Sebille (J. Combin. Des. 7 (1999) 107–112) proved that if, for some positive integers, there exists an S_λ(t,k,v) such that

where m=min{s,v−k} and n=min{i,t}, then there exists an

having pairwise disjoint subdesigns S_λ(t,k,v). The purpose of this paper is to generalize these two constructions in order to produce a new recursive construction of t-designs and a new extension theorem of t-designs.     

Explicit and exact special solutions for BBM-like B(m, n) equations with fully nonlinear dispersion

In this paper, the BBM-like equations with fully nonlinear dispersion, B(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxt = 0 which exhibit solutions with compact support and with solitary patterns, are studied. The exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of the equations are found by a new method. The special cases, B(2, 2) and B(3, 3), are used to illustrate the concrete scheme of our approach presented by this paper in B(m, n) equations. The nonlinear equations B(m, n) are addressed for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of B(m, n) equations are established.     

Every tree is 3-equitable

A labeling of a graph is a function f from the vertex set to some subset of the natural numbers. The image of a vertex is called its label. We assign the label |f(u)−f(v)| to the edge incident with vertices u and v. In a k-equitable labeling the image of f is the set {0,1,2,…,k−1}. We require both the vertex labels and the edge labels to be as equally distributed as possible, i.e., if v_i denotes the number of vertices labeled i and e_i denotes the number of edges labeled i, we require |v_i−v_j|1 and |e_i−e_j|1 for every i,j in {0,1,2,…,k−1}. Equitable graph labelings were introduced by I. Cahit as a generalization for graceful labeling. We prove that every tree is 3-equitable.     

Small embeddings for partial cycle systems of odd length

We prove that if m is odd then a partial m-cycle system on n vertices can be embedded in an m-cycle system on at most m((m − 2)n(n − 1) + 2n + 1) vertices and that a partial weak Steiner m-cycle system on n vertices can be embedded in an m-cycle system on m(2n + 1) vertices.     

Loopless generation of up–down permutations

An up–down permutation P=(p₁,p₂,…,p_n) is a permutation of the integers 1 to n which satisfies constraints specified by a sequence C=(c₁,c₂,…,c_n−1) of U's and D's of length n−1. If c_i is U then p_i<p_i+1 otherwise p_i−1>p_i. A loopless algorithm is developed for generating all the up–down permutations satisfying any sequence C. Ranking and unranking algorithms are discussed.     

Near vector spaces over GF(q) and (v,q+1,1)-BIBD''s

The usual construction of (v,q+1,1)−BIBD's from vector spaces over GF(q) is generalized to the class of near vector spaces over GF(q). It is shown that every (v,q+1,1)−BIBD can be constructed from a near vector space over GF(q). Some corollaries are: Given a (v₁,q+1,1)−BIBD P₁,B₁ and a (v₂,q+1,1)−BIBD P₂,B₂, there is a ((q−1)v₁v₂+v₁+v₂,q+1,1)−BIBD P₃,B₃ containing P₁,B₁ and P₂,B₂ as disjoint subdesigns. If there is a (v,q+1,1)−BIBD then there is a ((q−1)v+1,q,1)−BIBD. Every finite partial (v,q,1)−BIBD can be embedded in a finite (v′,q+1,1)−BIBD.     

The coordinate representation of a graph and n-universal graph of radius 1

Every graph can be represented as the intersection graph on a family of closed unit cubes in Euclidean space Eⁿ. Cube vertices have integer coordinates. The coordinate matrix, A(G)={v_nk} of a graph G is defined by the set of cube coordinates. The imbedded dimension of a graph, Bp(G), is a number of columns in matrix A(G) such that each of them has at least two distinct elements v_nk≠v_pk. We show that Bp(G)=cub(G) for some graphs, and Bp(G)n−2 for any graph G on n vertices. The coordinate matrix uses to obtain the graph U of radius 1 with 3ⁿ⁻² vertices that contains as an induced subgraph a copy of any graph on n vertices.     

The DFS-heuristic for orthogonal graph drawing

In this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of m−2 bends.