Disjoint Paths Of Length Research Articles

The Wide Diameter and Fault Diameter of Exchanged Crossed Cube

The fault diameter and wide diameter are commonly used to measure the fault tolerance and transmission delay of interconnection networks beyond traditional diameter. The [Formula: see text]-wide diameter of graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there exist at least [Formula: see text] internally vertex disjoint paths of length at most [Formula: see text] for any two distinct vertices in [Formula: see text]. The [Formula: see text]-fault diameter of graph [Formula: see text], denoted by [Formula: see text], is the maximum diameter of the survival graph obtained by deleting at most [Formula: see text] vertices in [Formula: see text]. The exchanged crossed cube, as a compounded interconnection network denoted by [Formula: see text], holds the desirable properties of both crossed cube and exchanged hypercube, while achieving a better balanced between cost and performance of the parallel computing systems. In this paper, we construct [Formula: see text] internally vertex disjoint paths between any two distinct vertices of [Formula: see text]. Moreover, we determine the upper and lower bounds of [Formula: see text]-wide diameter and [Formula: see text]-fault diameter of [Formula: see text], i.e., [Formula: see text], which shows that the exchanged crossed cube has better efficiency and reliability than that of the exchanged hypercube.

Hamilton‐connected {claw, bull}‐free graphs

AbstractThe generalized bull is the graph obtained by attaching endvertices of two disjoint paths of lengths to two vertices of a triangle. We prove that every 3‐connected ‐free graph, where , is Hamilton‐connected. The results are sharp and complete the characterization of forbidden induced bulls implying Hamilton‐connectedness of a 3‐connected {claw, bull}‐free graph.

Hamiltonian line graphs with local degree conditions

Let N1,1,1 be the graph formed by attaching a pendant edge to each vertex of a triangle, and B1,2 be a graph obtained by attaching end vertices of two disjoint paths of lengths 1,2 to two vertices of a triangle. Broersma (1993) [2] and Čada et al. (2016) [3] conjectured that for a 2-connected claw-free simple graph G and for a fixed graph Γ∈{N1,1,1,B1,2,P6}, if δΓ(G)=min⁡{dG(v):dH(v)=1 for any induced subgraph H≅Γ in G}≥|V(G)|−23, then G is Hamiltonian. While Chen settles this conjecture recently, the following two results of the conjecture for 3-connected line graphs are proved.(i) For real numbers a,b with 0<a<1, there exists a family F(a,b) of finitely many nonsupereulerian graphs, such that for any 3-connected line graph H=L(G) of a simple graph G, if δN1,1,1(H)≥a|V(H)|+b, then either H is Hamiltonian or G is contractible to a member in F(a,b).(ii) Let H=L(G) be a 3-connected line graph of a simple graph G with |V(H)|≥116. If δN1,1,1(H)≥|V(H)|+510, then either H is Hamiltonian or G is isomorphic to the graph P(10)′, which is formed from the Petersen graph P(10) by attaching |V(H)|−1510 pendant edges to every vertex of P(10).

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

(Theta, triangle)‐free and (even hole, K4)‐free graphs—Part 1: Layered wheels

AbstractWe present a construction called layered wheel. Layered wheels are graphs of arbitrarily large treewidth and girth. They might be an outcome for a possible theorem characterizing graphs with large treewidth in terms of their induced subgraphs (while such a characterization is well‐understood in terms of minors). They also provide examples of graphs of large treewidth and large rankwidth in well‐studied classes, such as (theta, triangle)‐free graphs and even‐hole‐free graphs with no (where a hole is a chordless cycle of length at least four, a theta is a graph made of three internally vertex disjoint paths of length at least two linking two vertices, and is the complete graph on four vertices).

Edge decomposition of connected claw-free cubic graphs

It was conjectured by Hoffmann–Ostenhof that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a family of disjoint cycles. In this paper, we show that the conjecture is true for connected claw-free cubic graphs, and, furthermore, any edge not contained any triangle appears only on the tree or on the matching. Then we show that the edge set of every connected cubic graph (except for K4 and K3,3) can be decomposed into a spanning tree and a family of disjoint paths of length at most 2.

Spectra, Hitting Times and Resistance Distances ofq- Subdivision Graphs

AbstractSubdivision, triangulation, Kronecker product, corona product and many other graph operations or products play an important role in complex networks. In this paper, we study the properties of $q$-subdivision graphs, which have been applied to model complex networks. For a simple connected graph $G$, its $q$-subdivision graph $S_q(G)$ is obtained from $G$ through replacing every edge $uv$ in $G$ by $q$ disjoint paths of length 2, with each path having $u$ and $v$ as its ends. We derive explicit formulas for many quantities of $S_q(G)$ in terms of those corresponding to $G$, including the eigenvalues and eigenvectors of normalized adjacency matrix, two-node hitting time, Kemeny constant, two-node resistance distance, Kirchhoff index, additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index. We also study the properties of the iterated $q$-subdivision graphs, based on which we obtain the closed-form expressions for a family of hierarchical lattices, which has been used to describe scale-free fractal networks.

Improved approximation algorithms for weighted 2-path partitions

We investigate two NP-complete vertex partition problems on edge-weighted complete graphs with 3k vertices. The first problem asks to partition the graph into k vertex disjoint paths of length 2 (referred to as 2-paths) such that the total weight of the paths is maximized. We present a cubic time approximation algorithm that computes a 2-path partition whose total weight is at least .5833 of the weight of an optimal partition, improving upon the (.5265−ϵ)-approximation algorithm of Tanahashi and Chen (2010). Restricting the input to graphs with edge weights in {0,1}, we present a .75 approximation algorithm improving upon the .55-approximation algorithm of Hassin and Schneider (2013).Combining this algorithm with a previously known approximation algorithm for the 3-Set Packing problem, we obtain a .6-approximation algorithm for the problem of partitioning a {0,1}-edge-weighted graph into k vertex disjoint triangles of maximum total weight. The best known approximation algorithm for general weights is due to Chen and Tanahashi (2009) and achieves an approximation ratio of .5257.

Cycles with two blocks in k‐chromatic digraphs

AbstractLet k and ℓ be positive integers. A cycle with two blocks is a digraph obtained by an orientation of an undirected cycle, which consists of two internally (vertex) disjoint paths of lengths at least k and ℓ, respectively, from a vertex to another one. A problem of Addario‐Berry, Havet and Thomassé [J. Combin. Theory Ser. B 97 (2007), 620–626] asked if, given positive integers k and ℓ such that , any strongly connected digraph D containing no has chromatic number at most . In this article, we show that such digraph D has chromatic number at most , improving the previous upper bound of Cohen et al. [Subdivisions of oriented cycles in digraphs with large chromatic number, to appear]. We also show that if in addition D is Hamiltonian, then its underlying simple graph is ‐degenerate and thus the chromatic number of D is at most , which is tight.

The fault-diameter and wide-diameter of twisted hypercubes

The ℓ-fault-diameter of a graph G is the minimum d such that the diameter of G−X is at most d for any subset X of V(G) of size |X|≤ℓ−1. The ℓ-wide-diameter of G is the minimum integer d for which there exist at least ℓ internally vertex disjoint paths of length at most d between any two distinct vertices in G. These two parameters measure the fault tolerance and transmission delay in communication networks modelled by graphs. Twisted hypercubes are variation of hypercubes defined recursively: K1 is the only 0-dimension twisted hypercube, and for n≥1, an n-dimensional twisted hypercube Gn is obtained from the disjoint union of two (n−1)-dimensional twisted hypercubes Gn−1′ and Gn−1′′ by adding a perfect matching between V(Gn−1′) and V(Gn−1′′). Recently, two types of twisted hypercubes Zn,k and Hn are introduced in Zhu (2017). This paper gives an upper bound for the n-fault-diameters of special families of twisted hypercubes of dimension n. As a consequence of this result, the n-fault-diameter of Hn is (1+o(1))nlog2n. For k=⌈log2n−2log2log2n⌉, the n-fault-diameter of Zn,k is also (1+o(1))nlog2n. This bound is asymptotically optimal, as any n-dimensional twisted hypercube has diameter greater than nlog2n. Then we extend a result in Qi and Zhu (2017) about Zn,k to Hn and prove that the κ(n)-wide-diameter of Hn is (1+o(1))nlog2n.

Four forbidden subgraph pairs for hamiltonicity of 3-connected graphs

For non-negative integers i, j and k, we denote the generalized net as N i,j,k , which is a triangle with disjoint paths of length i, j and k, attached to distinct vertices of the triangle. In this paper, we prove that every 3-connected {K 1,3,N 8−i,i,1}-free graph is hamiltonian, where 1 ≤ i ≤ 4.

Every 3-connected $$\{K_{1,3},N_{1,2,3}\}$$ { K 1 , 3 , N 1 , 2 , 3 } -free graph is Hamilton-connected

A graph G is $$\{X,Y\}$${X,Y}-free if it contains neither X nor Y as an induced subgraph. Pairs of connected graphs X, Y such that every 3-connected $$\{X,Y\}$${X,Y}-free graph is Hamilton-connected have been investigated recently in (2002, 2000, 2012). In this paper, it is shown that every 3-connected $$\{K_{1,3},N_{1,2,3}\}$${K1,3,N1,2,3}-free graph is Hamilton-connected, where $$N_{1,2,3}$$N1,2,3 is the graph obtained by identifying end vertices of three disjoint paths of lengths 1, 2, 3 to the vertices of a triangle.

Modular Multiplicative Divisor Labeling of Some Path Related Graphs

A graph G (V,E) with |V| = n is said to have modular multiplicative divisor labeling if there exist a bijection f: V(G) → {1, 2, …,} and the induced function f * : E(G) → {0, 1, 2, …, n - 1} where f(uv) = f(u)f(v) (mod n) such that n divides the sum of all edge labels of G. We prove that the path P n , and the graph P a, b (a graph which connects two vertices by means of "b" internally disjoint paths of length "a" each), shadow graph of a path and the cartesian product P n × P 1 , (n is not a multiple of 6) admits modular multiplicative divisor labeling. Also we discuss the upper bound for the number of edges in a modular multiplicative divisor graphs. AMS Subject Classification: 05C78

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

For non-negative integers i, j and k, let Ni,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 {K1,3,N3,3,3}-free graph is Hamiltonian. This result is sharp in the sense that for any integer i > 3, there exist infinitely many 3-connected {K1,3,Ni,3,3}-free non-Hamiltonian graphs.

Two Forbidden Subgraph Pairs for Hamiltonicity of 3-Connected Graphs

For integers i, j, k with $${i\geq j\geq k\geq 0}$$ , let N i, j, k be the graph obtained by identifying end vertices of three disjoint paths of lengths i, j, k to the vertices of a triangle. In this paper, we show that every 3-connected {K 1,3, N i, 7-i, 2}-free graph is hamiltonian, where $${i \in \{4,5\}}$$ . This result is sharp in the sense that no one of the numbers i, 7?i and 2 in N i, 7-i, 2 can be replaced by a larger number.

The maximum flow problem with disjunctive constraints

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly $\mathcal{NP}$ -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly $\mathcal{NP}$ -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.

Closure and forbidden pairs for 2-factors

Pairs of connected graphs X , Y such that a graph G is 2-connected and X Y -free implies that G is hamiltonian were characterized by Bedrossian. Using the closure concept for claw-free graphs, the first author simplified the characterization by showing that if considering the closure of G , the list in the Bedrossian characterization can be reduced to one pair, namely, K 1 , 3 , N 1 , 1 , 1 (where K i , j is the complete bipartite graph, and N i , j , k is the graph obtained by identifying endvertices of three disjoint paths of lengths i , j , k to the vertices of a triangle). Faudree et al. characterized pairs X , Y such that G is 2-connected and X Y -free implies that G has a 2-factor. Recently, the first author et al. strengthened the closure concept for claw-free graphs such that the closure of a graph has stronger properties while still preserving the (non)-existence of a 2-factor. In this paper we show that, using the 2-factor closure, the list of forbidden pairs for 2-factors can be reduced to two pairs, namely, K 1 , 4 , P 4 and K 1 , 3 , N 1 , 1 , 3 .

Wide diameters of de Bruijn graphs

The wide diameter of a graph is an important parameter to measure fault-tolerance of interconnection network. This paper proves that for any two vertices in de Bruijn undirected graph UB(d,n), there are 2d−2 internally disjoint paths of length at most 2n+1. Therefore, the (2d−2)-wide diameter of UB(d,n) is not greater than 2n+1.

Integer programming formulations for the two 4‐hop‐constrained paths problem

AbstractIn this article, we consider the two 4‐hop‐constrained paths problem, which consists, given a graph G = (N,E) and two nodes s,t ∈ N, of finding a minimum cost subgraph in G containing at least two node‐ (resp., edge‐) disjoint paths of length at most 4 between s and t. We give integer programming formulations, in the space of the design variables, for both the node and edge versions of this problem. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 49(2), 135–144 2007

Construction of asymmetric connectors of depth two

An ( n , N ) -connector of depth d is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length at most d joining each input to its corresponding output. In this paper we consider the problem of construction of sparse depth two connectors with n ≪ N . We use posets of star products and their matching properties to construct such connectors. In particular, this gives a simple explicit construction for connectors of size O ( N log n / log log n ) . Thus our earlier idea to use other posets than the family of subsets of a finite set was successful.