On the minimal reducible bound for outerplanar and planar graphs

Let L be the set of all additive and hereditary properties of graphs. For P₁, P₂ L we define the reducible property R = P₁ P₂ as follows: G P₁P₂ if there is a bipartition (V₁, V₂) of V(G) such that V₁ P₁ and V₂ P₂. For a property P L, a reducible property R is called a minimal reducible bound for P if P R and for each reducible property R′, R′ R → P R′. It is proved that the class of all outerplanar graphs has exactly two minimal reducible bounds in L. Some related problems for planar graphs are discussed.     

On the existence of a combinatorial Schlegel diagram of a simplical unstacked 3-polytope with a prescribed set of vertices

It is shown that, except for two well defined configurations, any finite set with exactly three points on ∂V is the vertex set of a triangulation of [V] which is combinatorially the Schlegel diagram of a simplicial unstacked 3-polytope.     

Class Steiner trees and VLSI-design

We consider a generalized version of the Steiner problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with required classes of vertices and Steiner vertices, find a shortest connected subgraph containing at least one vertex of each required class. We show that this problem is NP-hard, even if there are no Steiner vertices and the graph is a tree. Moreover, the same complexity result holds if the input class Steiner graph additionally is embedded in a unit grid, if each vertex has degree at most three, and each class consists of no more than three vertices. For similar restricted versions, we prove MAX SNP-hardness and we show that there exists no polynomial-time approximation algorithm with a constant bound on the relative error, unless P = NP. We propose two efficient heuristics computing different approximate solutions in time O(¦E¦+¦V¦log¦V¦) and in time O(c(¦E¦+¦V¦log¦V¦)), respectively, where E is the set of edges in the given graph, V is the set of vertices, and c is the number of classes. We present some promising implementation results. kw]Steiner Tree; Heuristic; Approximation complexity; MAX-SNP-hardness     

Quaternionic Numerical Range and Real Subspaces

Let W(A) be the numerical range of an n × n quaternionic matrix A and V a real subspace of the skew field of real quaternions. In this note the authors consider the relation among the shape of W(A), the convexity of V∩W(A): and the validity of the equality V∩W(A) = W_v(A), where W_v (A) is the orthogonal projection of W(A) into V.     

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.     

The k-ball l-path branch weight centroid

If x is a vertex of a tree T of radius r, if k and l are integers, if 0 k r, 0 l r, and if P is an l-path with one end at x, then define β(x; k, P) to be the number of vertices of T that are reachable from x via the l-path P and that are outside of the k-ball about x. That is, β(x;k,P) = {yεV(T):y is reachable from x via P,d(x,y) > k}. Define the k-ball l-path branch weight of x, denoted β(x;k,l), to be max {β(x;k,P):P an l-path with one end at x}, and define the k-balll-path branch weight centroid of T, denoted B(T;k,l), to be the set xεV(T): β(x;k,l) β(y;k,l), yεV(T). This two-parameter family of central sets in T includes the one-parameter family of central sets called the k-nuclei introduced by Slater (1981) which has been shown to be the one parameter family of central sets called the k-branch weight centroids by Zaw Win (1993). It also includes the one-parameter family of central sets called the k-ball branch weight centroid introduced by Reid (1991). In particular, this new family contains the classical central sets, the center and the median (which Zelinka (1968) showed is the ordinary branch weight centroid). The sets obtained for particular values of k and l are examined, and it is shown that for many values they consist of one vertex or two adjacent vertices.     

Multicolor routing in the undirected hypercube

An undirected routing problem is a pair (G,R) where G is an undirected graph and R is an undirected multigraph such that V(G)=V(R). A solution to an undirected routing problem (G,R) is a collection P of undirected paths of G (possibly containing multiple occurrences of the same path) such that edges of R are in one-to-one correspondence with the paths of P, with the path corresponding to edge {u,v} connecting u and v. We say that a collection of paths P is k-colorable if each path of P can be colored by one of the k colors so that the paths of the same color are edge-disjoint (each edge of G appears at most once in the paths of each single color). In the circuit-switched routing context, and in optical network applications in particular, it is desirable to find a solution to a routing problem that is colorable with as few colors as possible. Let Q_n denote the n-dimensional hypercube, for arbitrary n1. We show that a routing problem (Q_n,R) always admits a 4d-colorable solution where d is the maximum vertex degree of R. This improves over the 16d/2-color result which is implicit in the previous work of Aumann and Rabani [SODA95, pp. 567–576]. Since, for any positive d, there is a multigraph R of degree d such that any solution to (Q_n,R) requires at least d colors, our result is tight up to a factor of four. In fact, when d=1, it is tight up to a factor of two, since there is a graph of degree one (the antipodal matching) that requires two colors.     

On total covers of graphs

A total cover of a graph G is a subset of V(G)E(G) which covers all elements of V(G)E(G). The total covering number ₂(G) of a graph G is the minimum cardinality of a total cover in G. In [1], it is proven that ₂(G)[n/2] for a connected graph G of order n. Here we consider the extremal case and give some properties of connected graphs which have a total covering number [n/2]. We prove that such a graph with even order has a 1-factor and such a graph with odd order is factor-critical.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

The complexity of irredundant sets parameterized by size

An irredundant set of vertices V′V in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size n−k is fixed-parameter tractable.     

Pseudo-Hamiltonian-connected graphs

Given a graph G and a positive integer k, denote by G[k] the graph obtained from G by replacing each vertex of G with an independent set of size k. A graph G is called pseudo-k Hamiltonian-connected if G[k] is Hamiltonian-connected, i.e., every two distinct vertices of G[k] are connected by a Hamiltonian path. A graph G is called pseudo Hamiltonian-connected if it is pseudo-k Hamiltonian-connected for some positive integer k. This paper proves that a graph G is pseudo-Hamiltonian-connected if and only if for every non-empty proper subset X of V(G), |N(X)|>|X|. The proof of the characterization also provides a polynomial-time algorithm that decides whether or not a given graph is pseudo-Hamiltonian-connected. The characterization of pseudo-Hamiltonian-connected graphs also answers a question of Richard Nowakowski, which motivated this paper.     

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.     

A class of infinitely divisible variance functions with an application to the polynomial case

Let be a natural exponential family on ??? with variance function (V, Ω). Here, Ω is the mean domain of and V is its variance expressed in terms of the mean μ ε Ω. In this note we prove the following result. Consider an open interval Ω = (0, b), 0 < b ∞, and a positive real analytic function V on Ω. If V² is absolutely monotone on [0, b) and V has the form μt(μ), where 1 and t is real analytic in a neighborhood of zero, then there exits an infinitely divisible natural exponential family with variance function (V, Ω). We illustrate this result with several examples of general nature.     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

路的字典积的邻和可区别边染色

图G的正常[k]-边染色σ是指颜色集合为[k]={1,2,…,k}的G的一个正常边染色.用w_σ（x）表示顶点x关联边的颜色之和,即w_σ（x）=∑_e∋x σ（e）,并称w_σ（x）关于σ的权.图G的k-邻和可区别边染色是指相邻顶点具有不同权的正常[k]-边染色,最小的k值称为G的邻和可区别边色数,记为χ'_∑（G）.现得到了路P_n与简单连通图H的字典积P_n[H]的邻和可区别边色数的精确值,其中H分别为正则第一类图、路、完全图的补图.     

On a conjecture of C. Johnson

Given a pair of n×n matricesA and B, one may form a polynomial P(A,B,λ) which generalizes the characteristic polynomial of BP(B,λ). In particular, when A=I (identity), P(A, B,λ) = P(B,λ), the characteristic polynomial of B. C. Johnson has conjectured [1] (among other things) that when A and B are hermitian and A is positive definite, then P(A,B,λ) has real roots. The case n=2 can be done by hand. In this paper we verify the conjecture for n=3.     

The countability index of the endomorphism semigroup of a vector space

The countability index C(S) of a semigroup S is the least positive integer n, if such an integer exists, with the property that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists. C(S) is defined to be infinite. Let V be a vector space over a field F and denote by End V the endomorphism semigroup of V. In the two main results, it is determined precisely when C(End V)=2 and when C(End V)=x SpecificallyC(End V)=2 if and only if V is infinite dimensional or dim V=1 and F is finite and C(End V)=x if and only if F is infinite and dim V is an integer N≥1.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)∈F[x].LetxV → V be a linear operator. Let S_φbe the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S _φ to be a subspace.