8564003
Ramsey Problems Involving Degrees in Edge-colored Complete Graphs of Vertices Belonging to Monochromatic Subgraphs
We consider monochromatic subragraphs in two-colored graphs as guaranteed by Ramsey's theorem, and ask various questions concerning the degree in the two-colored complete graphs of vertices which are part of these subgraphs.
9764979
Tough enough chordal graphs are Hamiltonian
3811952
Degree Sequences with Repeated Values
A given nonincreasing sequence D = (d1, d2, · · · , dn) is said to contain a (nonincreasing) repetition sequence D∗ = (di1 , di2 , · · · , dik ) for some k ≤ n − 2 if all values of D −D∗ are distinct and for any dil ∈ D ∗ there exists some dt ∈ D −D∗ such that dil = dt. For any pair of integers n and k with n ≥ k + 2, we investigate the existence of a graphic sequence which contains a given repetition sequence. Our main theorem contains the known results for the special case di1 = dik if k = 1 or k = 2 (see [1, 5, 2]). 1supported in part by a foundation of Academia Sinica.
39660968
A special k-coloring for a connected k-chromatic graph
For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number ?(G) = k can be properly vertex colored by k colors so that for each pair of vertices xo and xp in any color class there exist vertices x1, x2, ?, xp-1 of the same class with dist(xi, xi+1) ? f(k) for each i, 0 ? i ? p ? 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.We prove that f(k) < 12k when the order of the graph is ? k(k ? 2) + 1.
1039631
New Conditions for k-ordered Hamiltonian Graphs
We show that in any graph G on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y, we can fix the order of k vertices on a given cycle and find a hamiltonian cycle encountering these vertices in the same order, as long as k < n/12 and G is d(k + 1)/2e-connected. Further we show that every b3k/2cconnected graph on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y is k-ordered hamiltonian, i.e. for every ordered set of k vertices we can find a hamiltonian cycle encountering these vertices in the given order. Both connectivity bounds are best possible.
43538481
Vertex-disjoint cycles containing specified vertices in a bipartite graph
A minimum degree condition is given for a bipartite graph to contain a 2-factor each component of which contains a previously specified vertex. 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004
5706395
Combinatorial Properties of Generalized Hypercube Graphs
This paper investigates combinatorial properties of generalized hypercube graphs including best containers, wide diameter, and fault diameter. These properties have received much attention recently in the study of interconnection networks.
45776383
Efficient Parallel Algorithms for Doubly Convex-Bipartite Graphs
Suppose that G = (S, T, E) is a bipartite graph. An ordering of S(T) has the adjacency property if for each vertex in T(S), its adjacent vertices in S(T) are consecutive in the ordering. If there exist orderings of S and T which have the adjacency property, G is called a doubly convex-bipartite graph. In this paper, a parallel algorithm is proposed to recognize a doubly convex-bipartite graph. The algorithm runs in O(log n) time using O(n3log n) processors on the CRCW PRAM, or O(log2 n) time using O(n3log2 n) processors on the CREW PRAM.
44397360
Reducing Time Complexities of Semigroup Computations on Mesh-Connected Computers with Multiple Broadcasting
10538824
Parallel Decomposition of Generalized Series-Parallel Graphs
Generalized series-parallel (GSP) graphs belong to the class of decomposable graphs which can be represented by their decomposition trees. Given a decomposition tree of a GSP graph, there are many graph-theoretic problems which can be solved efficiently. An efficient parallel algorithm for constructing a decomposition tree of a given GSP graph is presented. It takes O(log n) time with C(m, n) processors on a CRCW PRAM, where C(m, n) is the number of processors required to find connected components of a graph with in edges and n vertices in logarithmic time. Based on our algorithmic results, we also derive some properties for GSP graphs, which may be of interest in and of themselves.
17147705
Distributed algorithms for the quickest path problem
Let N = (V, A, C, L) be a network with node set V, arc set A, positive arc capacity function C, and nonnegative arc lead time function L. The quickest path problems is to find paths in N to transmit a given amount of data such that the transmission time is minimized. In this paper, distributed algorithms are developed for the quickest path problem in an asynchronous communication network. For the one-source quickest path problem, we present three algorithms that require O(rn2) messages and O(2) time, O(n) messages and O(rn) time, and O1+elog w) messages and O(rn1+elog w) time for any e, 0<e<1, respectively, where m = |A|, n = |V|, r is the number of distinct capacity values of N, and w is the maxima arc weight of N. For the all-pairs quickest path problem, we present an algorithm that requires O(mn) messages and O(m) time.