ContributionsAn algorithm for the Tutte polynomials of graphs of bounded treewidth

Let k be a fixed, positive integer. We give an algorithm which computes the Tutte polynomial of any graph G of treewidth at most k in time O(n^{2+7 log₂ c}), where c is twice the number of partitions of a set with 3k + 3 elements and n the number of vertices of G.     

Nowhere-zero 3-flows of highly connected graphs

Let G be a k-edge-connected graph of order n. If k4log₂ n then G has a nowhere-zero 3-flow.     

Unavoidable doubly connected large graphs

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs:

(1) P_n, a path on n vertices;

(2) K_1,n^s, the graph obtained from K_1,n by subdividing an edge once;

(3) K_2,ne, the graph obtained from K_2,n by deleting an edge;

(4) K_2,n⁺, the graph obtained from K_2,n by adding an edge between the two degree-n vertices x₁ and x₂, and a pendent edge at each x_i.

Two applications of this result are also discussed in the paper.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

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.     

Some Ore-type Results for Matching and Perfect Matching in <Emphasis Type="Italic">k</Emphasis>-uniform Hypergraphs

Let S₁ and S₂ be two (k-1)-subsets in a k-uniform hypergraph H. We call S₁ and S₂ strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S₁ ∩ e ≠∅ and S₂ ∩ e ≠∅ or e ⊆ S₁ ∪ S₂ or e ⊇ S₁ ∪ S₂, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k²-k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k-1)-sets equal to 2n-4(k-1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k-2)k. If the degree sum of any two middle independent (k-1)-subsets is larger than 2(d-1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k-1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.     

Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t_n], where we include n in the product. If p is prime, p|n, and (₂^p) > n, we prove that p is the minimum t_n. If no such prime exists, we prove t_n √5n when n> 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then t_n = 3p> 3 √n/2. For n(n + u) a square > n², we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g₂(n) be minimal such that a square can be formed as the product of distinct integers from [n, g₂(n)] so that no pair of consecutive integers is omitted. We prove that g₂(n) 3n − 3, and list or conjecture the values of g₂(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.     

Bipartite dimensions and bipartite degrees of graphs

A cover (bipartite) of a graph G is a family of complete bipartite subgraphs of G whose edges cover G's edges. G'sbipartite dimension d(G) is the minimum cardinality of a cover, and its bipartite degree η(G) is the minimum over all covers of the maximum number of covering members incident to a vertex. We prove that d(G) equals the Boolean interval dimension of the irreflexive complement of G, identify the 21 minimal forbidden induced subgraphs for d 2, and investigate the forbidden graphs for d n that have the fewest vertices. We note that for complete graphs, d(K_n) = [log₂n], η(K_n) = d(K_n) for n 16, and η(K_n) is unbounded. The list of minimal forbidden induced subgraphs for η 2 is infinite. We identify two infinite families in this list along with all members that have fewer than seven vertices.     

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.     

Partition of a directed bipartite graph into two directed cycles

Let D = (V₁, V₂; A) be a directed bipartite graph with |V₁| = |V₂| = n 2. Suppose that d_D(x) + d_D(y) 3n + 1 for all x ε V₁ and y ε V₂. Then D contains two vertex-disjoint directed cycles of lengths 2n₁ and 2n₂, respectively, for any positive integer partition n = n₁ + n₂. Moreover, the condition is sharp for even n and nearly sharp for odd n.     

Extending matchings in planar graphs V

A graph G on at least 2n + 2 vertices in n-extendable if every set of n independent edges extends to (i.e., is a subset of) a perfect matching in G. It is known that no planar graph is 3-extendable. In the present paper we continue to study 2-extendability in the plane. Suppose independent edges e₁ and e₂ are such that the removal of their endvertices leaves at least one odd component C_o. The subgraph G[V(C_o) V(e₁) V(e₂)] is called a generalized butterfly (or gbutterfly). Clearly, a 2-extendable graph can contain no gbutterfly. The converse, however, is false.

We improve upon a previous result by proving that if G is 4-connected, locally connected and planar with an even number of vertices and has no gbutterfly, it is 2-extendable. Sharpness with respect to the various hypotheses of this result is discussed.     

A sequential sorting network analogous to the batcher merge

We present a network of delay log₂N, whose comparators have only log₂N different lengths with maximum length N/2. This network is log-sequential in that it will sort N data items when they are passed through it log₂Ntimes. The design, which is related to the Batcher odd-even merge, is distinctly different from the first known example of a log-delay log-sequential network, due to Dowd, Perl, Rudolf, and Saks. It is quite probably the best possible sorting network.     

Neighborhood unions and cyclability of graphs

A graph G is said to be cyclable if for each orientation of G, there exists a set S of vertices such that reversing all the arcs of with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n36. In this paper, we show that if for any three independent vertices x₁, x₂ and x₃, |N(x₁)N(x₂)|+|N(x₂)N(x₃)|+|N(x₃)N(x₁)|2n+1, then G is cyclable.     

Embeddings of the base and bundle isomorphisms

Let π_i :E_i→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections s_i :M→E_i. Suppose that U is an open neighborhood of s₁(M) in E₁ and F :U→E₂ a smooth embedding so that π₂Fs₁ :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E₁ and the induced bundle f^*E₂ are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<kn we give examples of nonisomorphic oriented bundles E₁ and E₂ of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E₁ and f^*E₂ are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.     

On an optimal stopping problem of Gusein-Zade

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n→∞) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1−t^*)^r, where t^*≈0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule τ_r,n lets approximately t^*n arrivals go by and then stops ‘almost immediately’, in the sense that τ_r,n/n→t^* in probability as n→∞, r→∞     

The Hoffman number of a graph

For a connected graph G with n vertices, let {λ₁,λ₂,…,λ_r} be the set of distinct positive eigenvalues of the Laplacian matrix of G. The Hoffman number μ(G) of G is defined by μ(G)=λ₁λ₂…λ_r/n. In this paper, we study some properties and applications of the Hoffman number.     

Upper bounds for nearly optimal diagonal scaling of matrices

Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D₁ such that all diagonal elements of D₁A^TAD₁ are equal to one. It is shown in this paper that the condition number of AD₁ can be at least (n/2)^1/2 times the minimal one. Some questions for a further research are posed.     

On Mills's conjecture on matroids with many common bases

In this paper, we shall prove a conjecture of Mills: for two (k+1)-connected matroids whose symmetric difference between their collections of bases has size at most k, there is a matroid that is obtained from one of these matroids by relaxing n₁ circuit-hyperplanes and from the other by relaxing n₂ circuit-hyperplanes, where n₁ and n₂ are non-negative integers such that n₁+n₂k     

How to select a loser

N people select a loser by flipping coins. Recursively, the 0-party continues until the loser is found. Among other things, it is shown that this process stops on the average after about log₂ N steps. Nevertheless, this very plausible result requires rather advanced methods.     

Aspects of semiorders within interval orders

Let s_k(n) be the largest integer such that every n-point interval order with no antichain of more than k points includes an s_k(n)-point semiorder. When k = 1, s₁(n) = n since all interval orders with no two-point antichains are chains. Given (c₁,...,c₅) = (1, 2, 3, 4), it is shown that s₂(n) = c_n for n 4, s₃(n) = c_n for n 5, and for all positive n, s₂ (n+4) =s₂(n)+3, s₃(n+5) = s₃(n)+3. Hence s₂ has a repeating pattern of length 4 [1, 2, 3, 3; 4, 5, 6, 6; 7, 8, 9, 9;...], and s₃ has a repeating pattern of length 5 [1, 2, 3, 3, 4; 4, 5, 6, 6, 7; 7, 8, 9, 9, 10;...].

Let s(n) be the largest integer such that every n-point interval order includes an s(n)-point semiorder. It was proved previously that for even n from 4 to 14, and that s(17) = 9. We prove here that s(15) = s(16) = 9, so that s begins 1, 2, 3, 3, 4, 4,..., 8, 8, 9, 9, 9. Since s(n)/n→0, s cannot have a repeating pattern.