All trees are 1-embeddable and all except stars are 2-embeddable

A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))=. It is known that all trees T with n (≥2) vertices and T K_1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e};and that it is 2-embeddable if,for every pair e₁, e₂ of edges, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e₁, e₂}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K_1,n−1 are 2-embeddable. In a certain sense, this result is sharp.     

On the construction of most reliable networks

The construction of most reliable networks is investigated. In particular, the study of restricted edge connectivity shows that general Harary graphs are max λ–min m_i for all i=λ, λ+1,…,2λ−3. As a consequence, this implies that for each pair of positive integers n and e, there is a graph of n vertices and e edges that is max λ–min m_i for all i=λ,λ+1,…,2λ−3. General Harary graphs that are max λ–min m_i for all i=λ,λ+1,…,2λ−2 are also constructed.     

An edge version of the matrix-tree theorem and the wiener index

Let T be a tree on n vertices. The Laplacian matrix is L(T)=D(T)-A(T), where D(T) is the diagonal matrix of vertex degrees and A(T) is the adjacency matrix. A special case of the Matrix-Tree Theorem is that the adjugate of L(T) is the n-by-n matrix of l's. The (n-l)-square "edge version" of L(T)is K(T). The main result is a graph-theoretic interpretation of the entries of the adjugate of K(T). As an application, it is shown that the Wiener Index from chemistry is the trace of this adjugate.     

Incidence matrices, geometrical bases, combinatorial prebases and matroids

For a relation A (C × D), where C,D are two finite sets, and an ordering σ of C we construct a matroid M(σ) on the set D. For the relation A with the incidence matrix  we also define a geometrical basis with respect to F, where F is a subset of the set of all circuits of the column matroid on Â. Geometrical bases are certain bases of this column matroid. We establish connections between the bases of matroids M(σ) and the geometrical bases of A with respect to F. These connections give a combinatorial way of constructing bases of the column matroid on  using a subset F of its circuits.

We also consider a matroid M and the incidence relation between what we call the extended circuits of M and the bases of M. Applying the technique above we obtain the matroids M(σ) on the set of bases of the matroid M. In case of the incidence relation between vertices and edges of a graph this technique yields a unique matroid, the usual matroid of the graph.

Some particular relations are considered: a class of relations with a certain property (the T-property) and the relation of inclusion of chambers in simplices in an affine point configuration.     

On the bounded domination number of tournaments

In a simple digraph, a star of degree t is a union of t edges with a common tail. The k-domination number γ_k(G) of digraph G is the minimum number of stars of degree at most k needed to cover the vertex set. We prove that γ_k(T)=n/(k+1) when T is a tournament with n14k lg k vertices. This improves a result of Chen, Lu and West. We also give a short direct proof of the result of E. Szekeres and G. Szekeres that every n-vertex tournament is dominated by at most lg n−lglg n+2 vertices.     

On a problem concerning ordered colourings

Let G be a connected graph with v(G) 2 vertices and independence number (G). G is critical if for any edge e of G:

1. (i) (G − e) > (G), if e is not a cut edge of G, and

2. (ii) v(G_i) − (G_i) < v(G) − (G), I = 1, 2, if e is a cut edge and G₁, G₂ are the two components of G − e.

Recently, Katchalski et al. (1995) conjectured that: if G is a connected critical graph, then with equality possible if and only if G is a tree. In this paper we establish this conjecture.     

Profile minimization on triangulated triangles

Given graph G=(V,E) on n vertices, the profile minimization problem is to find a one-to-one function f:V→{1,2,…,n} such that ∑_vV(G){f(v)−min_xN[v] f(x)} is as small as possible, where N[v]={v}{x: x is adjacent to v} is the closed neighborhood of v in G. The trangulated triangle T_l is the graph whose vertices are the triples of non-negative integers summing to l, with an edge connecting two triples if they agree in one coordinate and differ by 1 in the other two coordinates. This paper provides a polynomial time algorithm to solve the profile minimization problem for trangulated triangles T_l with side-length l.     

Almost perfect matchings in random uniform hypergraphs

We consider the following model H_r(n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = (r − 1)n. Each r-subset of V × (_r−1^U) is chosen to be an edge of H ε H_r(n, p) with probability p = p(n), all choices being independent. It is shown that for every 0 < < 1 if P = (C ln n)/n^r−1 with C = C() sufficiently large, then almost surely every subset V₁ V of size | V₁ | = (1 − )n is matchable, that is, there exists a matching M in H such that every vertex of V₁ is contained in some edge of M.     

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.     

Decomposably non-negative matrices

Let A be an mn- by - mn symmetric matrix. Partition A into m²n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)^t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.     

Additive Maps Preserving Moore-Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_nF and S_nF be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_nF to S_nF (respectively, M_nF) preserving Moore-Penrose inverses of matrices are characterized. We first characterize all additive Moore-Penrose inverse preserving maps from S_nF to M_nF, and thereby, all additive Moore-Penrose inverse preserving maps from S_nF to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

On size, circumference and circuit removal in 3-connected matroids

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)|4r(M)−5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M)4 and M has a basis B such that Me is not 3-connected for all e in E(M)−B, then |E(M)|3r(M)−4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)|3r(M)−c(M).     

Coloring permutation graphs in parallel

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having specific key properties. We propose an efficient parallel algorithm which colors an n-node permutation graph in O(log² n) time using O(n²/log n) processors on the CREW PRAM model. Specifically, given a permutation π we construct a tree T^*[π], which we call coloring-permutation tree, using certain combinatorial properties of π. We show that the problem of coloring a permutation graph is equivalent to finding vertex levels in the coloring-permutation tree.     

Large eigenvalues of the laplacian

Let G be a simple graph on n vertices and let L=L(G) be the Laplacian matrix of G corresponding to some ordering of the vertices. It is known that λ≤n for any eigenvalue λ of L. In this note we characterize when n is an eigenvalue of L with multiplicity m.     

Covering hypercubes by isometric paths

An isometric path is merely any shortest path between two vertices. If the vertices of the hypercube Q_n are represented by the set of 0–1 vectors of length n, an isometric path is obtained by changing the coordinates of a vector one at a time, never changing the same coordinate more than once. The minimum number of isometric paths required to cover the vertices of Q_n is at least 2ⁿ/(n+1). We show that when n+1 is a power of 2, the lower bound is in fact the minimum. In doing so, we construct a family of disjoint isometric paths which can be used to find an upper bound for additional classes of hypercubes.     

Induced subgraphs of given sizes

We say (n, e) → (m, f), an (m, f) subgraph is forced, if every n-vertex graph of size e has an m-vertex spanned subgraph with f edges. For example, as Turán proved, (n,e)→(k,(^k₂)) for e> t_{k − 1}(n) and (n,e) (^k₂)), otherwise. We give a number of constructions showing that forced pairs are rare. Using tools of extremal graph theory we also show infinitely many positive cases. Several problems remain open.     

The cycle structure of regular multipartite tournaments

A multipartite tournament is an orientation of a complete multipartite graph. A tournament is a multipartite tournament, each partite set of which contains exactly one vertex. Alspach (Canad. Math. Bull. 10 (1967) 283) proved that every regular tournament is arc-pancyclic. Although all partite sets of a regular multipartite tournament have the same cardinality, Alspach's theorem is not valid for regular multipartite tournaments. In this paper, we prove that if the cardinality common to all partite sets of a regular n-partite (n3) tournament T is odd, then every arc of T is in a cycle that contains vertices from exactly m partite sets for all m{3,4,…,n}. This result extends Alspach's theorem for regular tournaments to regular multipartite tournaments. We also examine the structure of cycles through arcs in regular multipartite tournaments.     

A reversal index for tournament matrices

Given a tournament matrix T, its reversal indexi_R(T), is the minimum k such that the reversal of the orientation of k arcs in the directed graph associated with T results in a reducible matrix. We give a formula for i_R(T) in terms of the score vector of T which generalizes a simple criterion for a tournament matrix to be irreducible. We show that i_R(T)≤[(n-1)/2] for any tournament matrix T of order n, with equality holding if and only if T is regular or almost regular, according as n is odd or even. We construct, for each k between 1 and [(n-1)/2], a tournament matrix of order n whose reversal index is k. Finally, we suggest a few problems.     

How many maxima can there be?

Let C be a planar region. Choose n points p₁,,p_nI.I.D. from the uniform distribution over C. Let M^C_n be the number of these points that are maximal. If C is convex it is known that either E(M^C_n)=Θ(√n)> or E(M^C_n)=O(log n). In this paper we will show that, for general C, there is very little that can be said, a-priori, about E(M^C_n). More specifically we will show that if g is a member of a large class of functions then there is always a region C such that E(M^C_n)=Θ(g(n)). This class contains, for example, all monotically increasing functions of the form g(n)= nln^βn, where 0<<1 and β0. This class also contains nondecreasing functions like g(n)=ln^*n. The results in this paper remain valid in higher dimensions.     

A faster 2-approximation algorithm for the minmax p-traveling salesmen problem on a tree

Given an edge-weighted tree T and an integer p1, the minmax p-traveling salesmen problem on a tree T asks to find p tours such that the union of the p tours covers all the vertices. The objective is to minimize the maximum of length of the p tours. It is known that the problem is NP-hard and has a (2−2/(p+1))-approximation algorithm which runs in O(p^p−1n^p−1) time for a tree with n vertices. In this paper, we consider an extension of the problem in which the set of vertices to be covered now can be chosen as a subset S of vertices and weights to process vertices in S are also introduced in the tour length. For the problem, we give an approximation algorithm that has the same performance guarantee, but runs in O((p−1)!·n) time.