An improved linear edge bound for graph linkages

A graph is said to be k-linked if it has at least 2k vertices and for every sequence s₁,…,s_k,t₁,…,t_k of distinct vertices there exist disjoint paths P₁,…,P_k such that the ends of P_i are s_i and t_i. Bollobás and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn.     

Minimum degree conditions for -linked graphs

For a fixed multigraph H with vertices w₁,…,w_m, a graph G is H-linked if for every choice of vertices v₁,…,v_m in G, there exists a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i). This generalizes the notions of k-linked, k-connected, and k-ordered graphs.Given a connected multigraph H with k edges and minimum degree at least two and n7.5k, we determine the least integer d such that every n-vertex simple graph with minimum degree at least d is H-linked. This value D(H,n) appears to equal the least integer d^′ such that every n-vertex graph with minimum degree at least d^′ is b(H)-connected, where b(H) is the maximum number of edges in a bipartite subgraph of H.     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

A sufficient condition for a graph to be weakly k-linked

For a pair (s, t) of vertices of a graph G, let λ_G(s, t) denote the maximal number of edge-disjoint paths between s and t. Let (s₁, t₁), (s₂, t₂), (s₃, t₃) be pairs of vertices of G and k > 2. It is shown that if λ_G(s_i, t_i) ≥ 2k + 1 for each i = 1, 2, 3, then there exist 2k + 1 edge-disjoint paths such that one joins s₁ and t₁, another joins s₂ and t₂ and the others join s₃ and t₃. As a corollary, every (2k + 1)-edge-connected graph is weakly (k + 2)-linked for k ≥ 2, where a graph is weakly k-linked if for any k vertex pairs (s_i, t_i), 1 ≤ i ≤ k, there exist k edge-disjoint paths P₁, P₂,…, P_k such that P_i joins s_i and t_i for i = 1, 2,…, k.     

Linked graphs with restricted lengths

A graph G is k-linked if G has at least 2k vertices, and for every sequence x₁,x₂,…,x_k,y₁,y₂,…,y_k of distinct vertices, G contains k vertex-disjoint paths P₁,P₂,…,P_k such that P_i joins x_i and y_i for i=1,2,…,k. Moreover, the above defined k-linked graph G is modulo (m₁,m₂,…,m_k)-linked if, in addition, for any k-tuple (d₁,d₂,…,d_k) of natural numbers, the paths P₁,P₂,…,P_k can be chosen such that P_i has length d_i modulo m_i for i=1,2,…,k. Thomassen showed that, for each k-tuple (m₁,m₂,…,m_k) of odd positive integers, there exists a natural number f(m₁,m₂,…,m_k) such that every f(m₁,m₂,…,m_k)-connected graph is modulo (m₁,m₂,…,m_k)-linked. For m₁=m₂=…=m_k=2, he showed in another article that there exists a natural number g(2,k) such that every g(2,k)-connected graph G is modulo (2,2,…,2)-linked or there is XV(G) such that |X|4k−3 and G−X is a bipartite graph, where (2,2,…,2) is a k-tuple.We showed that f(m₁,m₂,…,m_k)max{14(m₁+m₂++m_k)−4k,6(m₁+m₂++m_k)−4k+36} for every k-tuple of odd positive integers. We then extend the result to allow some m_i be even integers. Let (m₁,m₂,…,m_k) be a k-tuple of natural numbers and ℓk such that m_i is odd for each i with ℓ+1ik. If G is 45(m₁+m₂++m_k)-connected, then either G has a vertex set X of order at most 2k+2ℓ−3+δ(m₁,…,m_ℓ) such that G−X is bipartite or G is modulo (2m₁,…,2m_ℓ,m_ℓ+1,…,m_k)-linked, where

Our results generalize several known results on parity-linked graphs.     

On a conjecture of J. Pelikán

There exist infinitely many finite sequences (a₁, …, a_n) (a_i {0, 1}) such that Φ_{i = 1}^{n − k} a_ia_{i + k} is odd for each k = 0, 1, …, n − 1.     

On shortest disjoint paths in planar graphs

For a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}, the k disjoint paths problem is to find k vertex-disjoint paths P₁,…,P_k, where P_i is a path from s_i to t_i for each i=1,…,k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P_i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k=2,3.Min-Sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time.Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3.     

Highly parity linked graphs

A graph G is k-linked if G has at least 2k vertices, and for any 2k vertices x ₁,x ₂, …, x _k ,y ₁,y ₂, …, y _k , G contains k pairwise disjoint paths P ₁, …, P _k such that P _i joins x _i and y _i for i = 1,2, …, k. We say that G is parity-k-linked if G is k-linked and, in addition, the paths P ₁, …, P _k can be chosen such that the parities of their length are prescribed. Thomassen [22] was the first to prove the existence of a function f(k) such that every f(k)-connected graph is parity-k-linked if the deletion of any 4k-3 vertices leaves a nonbipartite graph. In this paper, we will show that the above statement is still valid for 50k-connected graphs. This is the first result that connectivity which is a linear function of k guarantees the Erdős-Pósa type result for parity-k-linked graphs. Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research and by Inoue Research Award for Young Scientists.     

Theorems of large deviations in the multivariate invariance principle

Let B be a real separable Banach space with norm |ß|_B, X, X₁, X₂, … be a sequence of centered independent identically distributed random variables taking values in B. Let s_n = s_n(t), 0 ≤ t ≤ 1 be the random broken line such that s_n(0) = 0, s_n(k/n) = n^−1/2 Σ_i=1^k X_i for n = 1, 2, … and k = 1, …, n. Denote |s_n|_B = sup_{0 ≤ t ≤ 1} |s_n(t)|_B and assume that w(t), 0 ≤ t ≤ 1 is the Wiener process such that covariances of w(1) and X are equal. We show that under appropriate conditions P(|s_n|_B > r) = P(|w|_B > r)(1 + o(1)) and give estimates of the remainder term. The results are new already in the case of B having finite dimension.     

A sufficient condition for graphs to be weaklyk-linked

We consider graphs, which are finite, undirected, without loops and in which multiple edges are possible. For each natural numberk letg(k) be the smallest natural numbern, so that the following holds:LetG be ann-edge-connected graph and lets ₁,...,s _k,t ₁,...,t _k be vertices ofG. Then for everyi {1,..., k} there existsa pathP _i froms _i tot _i, so thatP ₁,...,P _k are pairwise edge-disjoint. We prove      

Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity

In this work we show that among all n-vertex graphs with edge or vertex connectivity k, the graph G=K_k(K₁+K_n−k−1), the join of K_k, the complete graph on k vertices, with the disjoint union of K₁ and K_n−k−1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum.     

Edge-disjoint paths and cycles in n-edge-connected graphs

We consider finite undirected loopless graphs G in which multiple edges are possible. For integers k,l ≥ 0 let g(k, l) be the minimal n ≥ 0 with the following property: If G is an n-edge-connected graph, s₁, ?,s_k, t₁, ?,t_k are vertices of G, and f₁, ?,f_l, g₁, ?,g_l, are pairwise distinct edges of G, then for each i = 1, ?, k there exists a path P_i in G, connecting s_i and t_i and for each i = 1, ?,l there exists a cycle C_i in G containing f_i and g_i such that P₁, ?,P_k, C₁, ?, C_l are pairwise edge-disjoint. We give upper and lower bounds for g(k, l).     

Stability conditions for systems of linear nonautonomous delay differential equations

Systems of linear nonautonomous delay differential equations are considered which are of the form y′_i(t) = ∑_{k = 1}ⁿ ∝₀^T b_ik(t, s) y_k(t − s) dη_ik(s) − c_i(t) y_i(t), where I = 1,…, n. Sufficient conditions are derived for both the asymptotic stability and the instability of the zero solution. The main result is found by a monotone technique using elementary methods only. Moreover, additional criteria are obtained by using the method of Lyapunov functionals.     

Strongly indexable graphs and applications

In 1990, Acharya and Hegde introduced the concept of strongly k-indexable graphs: A (p,q)-graph G=(V,E) is said to be strongly k-indexable if its vertices can be assigned distinct numbers 0,1,2,…,p−1 so that the values of the edges, obtained as the sums of the numbers assigned to their end vertices form an arithmetic progression k,k+1,k+2,…,k+(q−1). When k=1, a strongly k-indexable graph is simply called a strongly indexable graph. In this paper, we report some results on strongly k-indexable graphs and give an application of strongly k-indexable graphs to plane geometry, viz; construction of polygons of same internal angles and sides of distinct lengths.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

On Four-Connecting a Triconnected Graph

We consider the problem of finding a smallest set of edges whose addition four-connects a triconnected graph. This is a fundamental graph-theoretic problem that has applications in designing reliable networks and improving statistical database security. We present an O(n · α(m, n) + m)-time algorithm for four-connecting an undirected graph G that is triconnected by adding the smallest number of edges, where n and m are the number of vertices and edges in G, respectively, and α(m, n) is the inverse Ackermann function. This is the first polynomial time algorithm to solve this problem exactly.In deriving our algorithm, we present a new lower bound for the number of edges needed to four-connect a triconnected graph. The form of this lower bound is different from the form of the lower bound known for biconnectivity augmentation and triconnectivity augmentation. Our new lower bound applies for arbitrary k and gives a tighter lower bound than the one known earlier for the number of edges needed to k-connect a (k − 1)-connected graph. For k = 4, we show that this lower bound is tight by giving an efficient algorithm to find a set of edges whose size equals the new lower bound and whose addition four-connects the input triconnected graph.     

Complexity results for scheduling tasks with discrete starting times

Suppose that n independent tasks are to be scheduled without preemption on a set of identical parallel processors. Each task T_i requires a given execution time τ_i and it may be started for execution on any processor at any of its prescribed starting times s_i1, s_i2, …, s_iki, with k_i ≤ k for some fixed integer k. We first prove that the problem of finding a feasible schedule on a single processor is NP-complete in the strong sense even when τ_i ε {τ, τ′} and k_i ≤ 3 for 1 ≤ i ≤ n. The same problem is, however, shown to be solvable in O(n log n) time, provided s_iki − s_i1 < τ_i for 1 ≤ i ≤ n. We then show that the problem of finding a feasible schedule on an arbitrary number of processors is strongly NP-complete even when τ_i ε {τ, τ′}, k_i = 2 and s_i2 − s_i1 = δ < τ_i for 1 ≤ i ≤ n. Finally a special case with k_i = 2 and s_i2 − s_i1 = 1, 1 ≤ i ≤ n, of the above multiprocessor scheduling problem is shown to be solvable in polynomial time.     

Every 3-connected {K 1,3,N 3,3,3}-free graph is Hamiltonian

For non-negative integers i, j and k, let N _i,j,k be the graph obtained by identifying end vertices of three disjoint paths of lengths i, j and k to the vertices of a triangle. In this paper, we prove that every 3-connected {K _1,3,N _3,3,3}-free graph is Hamiltonian. This result is sharp in the sense that for any integer i > 3, there exist infinitely many 3-connected {K _1,3,N _i,3,3}-free non-Hamiltonian graphs.     

Highly linked graphs

A graph with at least 2k vertices is said to bek-linked if, for any choices ₁,...,s _k ,t ₁,...,t _k of 2k distinct vertices there are vertex disjoint pathsP ₁,...,P _k withP _i joinings _i tot _i , 1ik. Recently Robertson and Seymour [16] showed that a graphG isk-linked provided its vertex connectivityk(G) exceeds . We show here thatk(G)22k will do.     

Second-order linearity of the general signed-rank statistic

Let X₁,…, X_n be i.i.d. random variables symmetric about zero. Let R_i(t) be the rank of |X_i − tn^−1/2| among |X₁ − tn^−1/2|,…, |X_n − tn^−1/2| and T_n(t) = Σ_{i = 1}ⁿφ((n + 1)⁻¹R_i(t))sign(X_i − tn^−1/2). We show that there exists a sequence of random variables V_n such that sup_{0 ≤ t ≤ 1} |T_n(t) − T_n(0) − tV_n| → 0 in probability, as n → ∞. V_n is asymptotically normal.