New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Independent Trees and Branchings in Planar Multigraphs

 The following statement is proved: Let G be a finite directed or undirected planar multigraph and s be a vertex of G such that for each vertex x≠s of G, there are at least k pairwise openly disjoint paths in G from x to s where k∉{3,4,5} if G is directed. Then there exist k spanning trees T ₁, … ,T _k in G directed towards s if G is directed such that for each vertex x≠s of G, the k paths from x to s in T ₁, … ,T _k are pairwise openly disjoint. – The case where G is directed and k∈{3,4,5} remains open. Received: January 30, 1995 / Revised: October 7, 1996     

Self-converse Mendelsohn designs with block size 4t + 2

A Mendelsohn design MD(v, k, λ) is a pair (X, B) where X is a v-set together with a collection B of cyclic k-tuples from X such that each ordered pair from X, as adjacent entries, is contained in exactly λk-tuples of B. An MD(v, k, λ) is said to be self-converse, denoted by SCMD(v, k, λ) = (X, B, f), if there is an isomorphic mapping from (X, B) to (X, B⁻¹), where B⁻¹ = {B⁻¹ = 〈x_k, x_k−1, … x₂, x₁〉; B = 〈x₁, … ,x_k〉 ∈ B.}. The existence of SCMD(v, 3, λ) and SCMD(v, 4, 1) has been settled by us. In this article, we will investigate the existence of SCMD(v, 4t + 2, 1). In particular, when 2t + 1 is a prime power, the existence of SCMD(v, 4t + 2, 1) has been completely solved, which extends the existence results for MD(v, k, 1) as well. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 283–310, 1999     

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Extremal graphs in connectivity augmentation

Let A(n, k, t) denote the smallest integer e for which every k‐connected graph on n vertices can be made (k + t)‐connected by adding e new edges. We determine A(n, k, t) for all values of n, k, and t in the case of (directed and undirected) edge‐connectivity and also for directed vertex‐connectivity. For undirected vertex‐connectivity we determine A(n, k, 1) for all values of n and k. We also describe the graphs that attain the extremal values. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 179–193, 1999     

Betti Numbers of Path Ideals of Trees

Let R = k[x₁,…, x_n], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by I_t(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

Characterization and Transformation of Invariants

We consider difference equations of order k _n+k ≥ 2 of the form: y_n+k = f(yn,…,yn+k-1), n= 0,1,2,… where f: D ^k → D is a continuous function, and D?R. We develop a necessary and sufficient condition for the existence of a symmetric invariant I(x ₁,…,x_k ) ∈C^∞[D^k,D]. This condition will be used to construct invariants for linear and rational difference equations. Also, we investigate the transformation of invariants under invertible maps. We generalize and extend several results that have been obtained recently.     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

Hypersurfaces containing flat subspaces

Lek k be an infinite field and suppose m.i. and n are positive integers such that t m We study the subset of k[x ₁,x ₂, … x_m ] which consists of 0 and the homogeneous members t of f of k[x ₁,x ₂, … x_m ] of fixed degree n such that there exists homogeneous F ₁, F ₂, … F_t in k[x ₁,x ₂, … x_m ] of degree one and homogenous g ₁ g ₂, …g_t , in k[x ₁,x ₂, … x_m ] such that f(x) = F ₁(x)g ₁(x) + F ₂(x)g ₂(x) + … + F _t (x)g _t (x) for each x in k ^m. In case k is algebrarcally closed we are able to prove that this set is an algebraic variety. Consequently. if k is also of characteristic 0 then we are able to prove that certain collections of symmetric k-valued multilinear functions are algebraic varieties.     

The diameter of randomly perturbed digraphs and some applications

The central observation of this paper is that if εn random arcs are added to any n‐node strongly connected digraph with bounded degree then the resulting graph has diameter 𝒪(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL‐complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ∼ 𝔻_n,ε/n we obtain a “smoothed” instance. We show that, with high probability, a log‐space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL ⫅̸ almost‐L then no heuristic can recognize similarly perturbed instances of (s,t)‐connectivity. Property Testing: A digraph is called k‐linked if, for every choice of 2k distinct vertices s₁,…,s_k,t₁,…,t_k, the graph contains k vertex disjoint paths joining s_r to t_r for r = 1,…,k. Recognizing k‐linked digraphs is NP‐complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k‐linked graphs with high probability, and rejects all graphs that are at least εn arcs away from being k‐linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Multicommodity flows in graphs

Suppose that G is a graph, and (s_i,t_i) (1≤i≤k) are pairs of vertices; and that each edge has a integer-valued capacity (≥0), and that q_i≥0 (1≤i≤k) are integer-valued demands. When is there a flow for each i, between s_i and t_i and of value q_i, such that the total flow through each edge does not exceed its capacity? Ford and Fulkerson solved this when k=1, and Hu when k=2. We solve it for general values of k, when G is planar and can be drawn so that s₁,…, s_l, t₁, …, t_l,…,t_l are all on the boundary of a face and s_l+1, …,S_k, t_l+1,…,t_k are all on the boundary of the infinite face or when t₁=?=t_l and G is planar and can be drawn so that s_l+1,…,s_k, t₁,…,t_k are all on the boundary of the infinite face. This extends a theorem of Okamura and Seymour.     

Quasi‐randomness of graph balanced cut properties

Quasi‐random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi‐randomness of graphs. Let k ≥ 2 be a fixed integer, α₁,…,α_k be positive reals satisfying \begin{align*}\sum_{i} \alpha_i = 1\end{align*} and (α₁,…,α_k)≠(1/k,…,1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V ₁,…,V _k of size α₁n,…,α_kn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi‐random. However, the method of quasi‐random hypergraphs they used did not provide enough information to resolve the case (1/k,…,1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi‐random. Janson also posed the same question in his study of quasi‐randomness under the framework of graph limits. In this paper, we positively answer their question. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A determinantal version of the frobenius-könig theorem

Let F be a field and let {d ₁,…,d_k } be a set of independent indeterminates over F. Let A(d ₁,…,d_k ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d ₁,…,d_k }. We assume that no d ¹ appears twice in A(d ₁,…,d_k ). We show that if det A(d ₁,…,d_k ) = 0 then A(d ₁,…,d_k ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p.     

The degree sequence of a random graph. I. The models

We show that the joint distribution of the degrees of a random graph can be accurately approximated by several simpler models derived from a set of independent binomial distributions. On the one hand, we consider the distribution of degree sequences of random graphs with n vertices and ½m edges. For a wide range of values of m, this distribution is almost everywhere in close correspondence with the conditional distribution {(X₁,…,X_n) | ∑ X_i=m}, where X₁,…,X_n are independent random variables, each having the same binomial distribution as the degree of one vertex. We also consider random graphs with n vertices and edge probability p. For a wide range of functions p=p(n), the distribution of the degree sequence can be approximated by {(X₁,…,X>_n) | ∑ X_i is even}, where X₁,…,X_n are independent random variables each having the distribution Binom (n−1, p′), where p′ is itself a random variable with a particular truncated normal distribution. To facilitate computations, we demonstrate techniques by which statistics in this model can be inferred from those in a simple model of independent binomial random variables. Where they apply, the accuracy of our method is sufficient to determine asymptotically all probabilities greater than n^−k for any fixed k. In this first paper, we use the geometric mean of degrees as a tutorial example. In the second paper, we will determine the asymptotic distribution of the tth largest degree for all functions t=t(n) as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 97–117 (1997)     

On Hamiltonian Powers of Digraphs

 For a strongly connected digraph D, the k-th power D ^k of D is the digraph with the same set of vertices, a vertex x being joined to a vertex y in D ^k if the directed distance from x to y in D is less than or equal to k. It follows from a result of Ghouila-Houri that for every digraph D on n vertices and for every k≥n/2, D ^k is hamiltonian. In the paper we characterize these digraphs D of odd order whose (⌈n/2 ⌉−1)-th power is hamiltonian. Revised: June 13, 1997     

Independent trees in graphs

IfG is a finite undirected graph ands is a vertex ofG, then two spanning treesT ₁ andT ₂ inG are calleds — independent if for each vertexx inG the paths fromx tos inT ₁ andT ₂ are openly disjoint. It is known that the following statement is true fork3: IfG isk-connected, then there arek pairwises — independent spanning, trees inG. As a main result we show that this statement is also true fork=4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weaklys — independent spanning trees (i.e., the tree paths from a vertex tos are edge disjoint) and for directed graphs.     

New families of graceful banana trees

Consider a family of stars. Take a new vertex. Join one end-vertex of each star to this new vertex. The tree so obtained is known as abanana tree. It is proved that the banana trees corresponding to the family of stars

1. (K_1,1, K_1,2,…, K_{1,t ?1}, (α + l) K_1,t, K_{1,t + 1}, …, K_1,n), α ? 0
2. (2K_1,1, 2K_1,2,…, 2K_{1,t? 1}, (α + 2)K_1,t, 2K_{1,t + 1, …}, 2K_1,n), 0 ? α <t and
3. (3K_1,t, 3K_1,2, …, 3K_1,n) are graceful.

     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008