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 .     

Number of minimum vertex cuts in transitive graphs

Let G be a k-regular vertex transitive graph with connectivity κ(G)=k and let m_k(G) be the number of vertex cuts with k vertices. Define m(n,k)=min{m_k(G): GT_n,k}, where T_n,k denotes the set of all k-regular vertex transitive graphs on n vertices with κ(G)=k. In this paper, we determine the exact values of m(n,k).     

ContributionsOn optimal orientations of cartesian products of graphs (I)

For a graph G, let D(G) be the family of strong orientations of G, and define [ovbar|d] (G) = min[d(D) vb D ] D(G), where d(D) denotes the diameter of the digraph D. Let G × H denote the cartesian product of the graphs G and H. In this paper, we determine completely the values of and , except , where K_n, P_n and C_n denote the complete graph, path and cycle of order n, respectively.     

ContributionsTriangular embeddings of Kn−Km with unboundedly large m

The author has proposed methods of constructing index 2 and 3 current graphs generating triangular embeddings of graphs K_n−K_m with unboundedly large m (as n increases). As a result, triangular embeddings of graphs of many families of graphs K_n−K_m with unboundedly large m were constructed. The paper gives a survey of these results and a short explanation of the methods.     

Multicolored forests in complete bipartite graphs

If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a=(a₁,a₂,…,a_k), in which a_i denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph K_n,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of K_n,n into multicolored trees.     

Bondage number of strong product of two paths

The bondage number b(G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m≥2 and n≥2, b(P_m?P_n) = 7 - r(m) - r(n) if (r(m), r(n)) = (1, 1) or (3, 3), 6 - r(m) - r(n) otherwise, where r(t) is a function of positive integer t, defined as r(t) = 1 if t ≡ 1 (mod 3), r(t) = 2 if t ≡ 2 (mod 3), and r(t) = 3 if t ≡ 0 (mod 3).     

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.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H A → A, h a → h.a, be an action of H on a commutative local Noetherian kalgebra (A, m). We say that this action is linearizable if there exists a minimal system x₁, …, x_n of generators of the maximal ideal m such that h.x_i ε kx₁ + …+ kx_n for all h ε H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.     

Bounded rank of C-algebras

We introduce a concept of the bounded rank (with respect to a positive constant) for unital C^*-algebras as a modification of the usual real rank and present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given n and K>0 there exists a separable unital C^*-algebra Z_n^K such that every other separable unital C^*-algebra of bounded rank with respect to K at most n is a quotient of Z_n^K.     

The polytope of degree sequences of hypergraphs

Let D_n(r) denote the convex hull of degree sequences of simple r-uniform hypergraphs on the vertex set {1,2,…,n}. The polytope D_n(2) is a well-studied object. Its extreme points are the threshold sequences (i.e., degree sequences of threshold graphs) and its facets are given by the Erdös–Gallai inequalities. In this paper we study the polytopes D_n(r) and obtain some partial information. Our approach also yields new, simple proofs of some basic results on D_n(2). Our main results concern the extreme points and facets of D_n(r). We characterize adjacency of extreme points of D_n(r) and, in the case r=2, determine the distance between two given vertices in the graph of D_n(2). We give a characterization of when a linear inequality determines a facet of D_n(r) and use it to bound the sizes of the coefficients appearing in the facet defining inequalities; give a new short proof for the facets of D_n(2); find an explicit family of Erdös–Gallai type facets of D_n(r); and describe a simple lifting procedure that produces a facet of D_n+1(r) from one of D_n(r).     

Split and balanced colorings of complete graphs

An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain K_n in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph K_N has a coloring which is not (r, n)-split is denoted by ƒ_r(n). Balanced (r,n)-colorings are defined as edge r-colorings of KN such that every subset of [N/r] vertices contains a monochromatic K_n in all colors. Then g_r(n) is defined as the smallest N such that K_N has a balanced (r, n)-coloring. The definitions imply that f_r(n) g_r(n). The paper gives estimates and exact values of these functions for various choices of parameters.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

On ramsey numbers for large disjoint unions of graphs

Let be a fixed finite set of connected graphs. Results are given which, in principle, permit the Ramsey number r(G, H) to be evaluated exactly when G and H are sufficiently large disjoint unions of graphs taken from . Such evaluations are often possible in practice, as shown by several examples. For instance, when m and n are large, and mn,

r(mK_k, nK_l)=(k − 1)m+ln+r(K_k−1, K_l−1)−2.

     

Solution to an extremal problem on bigraphic pairs with a <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connected realization

Let S=(a₁,...,a_m; b₁,...,b_n), where a₁,...,a_m and b₁,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S=(a₁,...,a_m; b₁,...,b_n) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a₁,...,a_m and b₁,...,b_n are the degrees of the vertices in X and Y, respectively. Let Z₃ be the cyclic group of order 3. Define σ(Z₃, m, n) to be the minimum integer k such that every bigraphic pair S=(a₁,...,a_m; b₁,...,b_n) with a_m, b_n ≥ 2 and σ(S)=a₁ +... + a_m ≥ k has a Z₃-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z₃, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z₃, m, n) for m ≥ n ≥ 4.     

3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting—preprocess a set of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure—preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching— preprocess a set C of n convex fat polygons, so that the k objects intersecting a “not-too-large” query polygon can be reported efficiently, and (iv) bounded-size segment shooting—preprocess a set C as in (iii), so that the first object (if exists) hit by a “not-too-long” oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(λ_s(n)log³n), where s is the maximum number of intersections between the boundaries of the (xy-projections) of any pair of objects, and λ_s(n) is the maximum length of (n, s) Davenport-Schinzel sequences. The data structure for the fourth problem is of size O(λ_s(n)log²n). The query time in the first problem is O(log⁴n), the query time in the second and third problems is O(log³n + klog²n), and the query time in the fourth problem is O(log³n).

We also present a simple algorithm for computing a depth order for a set as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for , if exists, is a linear order of , such that, if K₁, K₂ and K₁ lies vertically above K₂, then K₁ precedes K₂.) Unlike the algorithm of Agarwal et al. (1995) that might output a false order when a depth order does not exist, the new algorithm is able to determine whether such an order exists, and it is often more efficient in practical situations than the former algorithm.     

An extremal problem on potentially Kr,s-graphic sequences

We consider a variation of a classical Turán-type extremal problem (F. Chung, R. Graham, Erd s on Graphs: His Legacy of Unsolved Problems, AK Peters Ltd., Wellesley, 1998, Chapter 3) as follows: Determine the smallest even integer σ(K_r,s,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂++d_nσ(K_r,s,n) is potentially K_r,s-graphic, where K_r,s is a r×s complete bipartite graph, i.e., π has a realization G containing K_r,s as its subgraph. In this paper, we first give sufficient conditions for a graphic sequence being potentially K_r,s-graphic, and then we determine σ(K_r,r,n) for r=3,4.     

Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form

where W is a classical Weyl group, X is a one-dimensional character of W, and d(π) is the descent statistic. This completes a picture which is known when W is the symmetric group S_n (the Weyl group A_n−1). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B_n(C_n) and D_n. The B_n, case uses sign-reversing involutions, while the D_n case follows from a result of independent interest relating statistics for all three groups.     

ContributionsChromatic Ramsey numbers

Suppose G is a graph. The chromatic Ramsey number r_c(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let M_n = min[r_c(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that M_n = (n − 1)² + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M₆ and M₇.