Warren E. Shreve

Implementation techniques for the vehicle routing problem

Abstract Six methods for implementing the widely used Clarke-Wright algorithm for the vehicle routing problem (VRP) are presented and compared. Fifty-five large test problems are used to compare the methods. The methods involve alternative ways to access adjacency information in both low and high density problems. The results clearly establish methods of choice for VRP problems with given characteristics.

Note on graphs without repeated cycle lengths

Fix any positive integer n. Let S be the set of all Steinhaus graphs of order n(n - 1)-2 + 1. The vertices for each graph in S are the first n(n - 1)-2 + 1 positive integers. Let I be the set of all labeled graphs of order n with vertices of the form i(i - 1)-2 + 1 for the first n positive integers i. This article shows that the function φ : S ➝ I that maps a Steinhaus graph to its induced subgraph is a bijection. Therefore, any graph of order n is isomorphic to an induced subgraph of a Steinhaus graph of order n(n - 1)-2 + 1. This considerably tightens a result of Brigham, Carrington, and Dutton in [Brigham, Carrington, & Dutton, Combin. Inform. System Sci. 17 (1992)], which showed that this could be done with a Steinhaus graph of order 2n-1.

A partition approach to Vizing's conjecture

In 1963, Vizing [Vichysl.Sistemy9 (1963), 30–43] conjectured that γ(G × H) ≥ γ(G)γ(H), where G × H denotes the cartesian product of graphs, and γ(G) is the domination number. In this paper we define the extraction number x(G) and we prove that P2(G) ≤ x(G), and γ(G × H) ≥ x(G)γ(H), where P2(G) is the 2-packing number of G. Though the equality x(G) = γ(G) is proven to hold in several classes of graphs, we construct an infinite family of graphs which do not satisfy this condition. Also, we show the following lower bound: γ(G × H) ≥ γ(G)P2(H) + P2(G)(γ(H) − P2(H)).

A New Game Chromatic Number

Consider the following two-person game on a graphG.Players I and II move alternatively to color a yet uncolored vertex ofGproperly using a pre-specified set of colors. Furthermore, Player II can only use the colors that have been used, unless he is forced to use a new color to guarantee that the graph is colored properly. The game ends when some player can no longer move. Player I wins if all vertices ofGare colored. Otherwise Player II wins. What is the minimal number?g*(G)of colors such that Player I has a winning strategy? This problem is motivated by the game chromatic number?g(G)introduced by Bodlaender and by the continued work of Faigle, Kern, Kierstead and Trotter. In this paper, we show that?g*(T) ?3for each treeT.We are also interested in determining the graphsGfor which?(G)= ?g*(G),as well as?g*(G)for thek-inductive graphs wherekis a fixed positive integer.

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.

Pancyclicity mod k of claw-free graphs and K 1,4 -free graphs

Abstract For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K 1,4 -free graph G with minimum degree δ ( G )⩾ k +3 is pancyclic mod k and every claw-free graph G with δ ( G )⩾ k +1 is pancyclic mod k , which confirms Thomassens conjecture (J. Graph Theory 7 (1983) 261–271) for claw-free graphs.

Boundary value problems for second order nonlinear matrix differential equations

TIhe paper [11] of Tomastik is concerned with oscillation of nonlinear matrix differential equations. In the present paper the concern is with the solution of boundary value problems for such equations. In fact it is shown here that a result similar to that of Bebernes [1] may be applied to obtain the existence of solutions of boundary value problems. In particular, it is proven that under certain conditions including A, B and F positive definite n x n matrices, the boundary value problem

A converse problem in matrix differential equations

where J is some interval. Here P(t) is a symmetric nxn matrix and 0 and / r e spectively the nxn zero and identity matrices. Also X* and tr X, denote, respectively, the transpose and the trace of X. In the scalar case it is well known (1) and (2) together imply p(t)=k9 a nonnegative constant. See [1] and the references therein. The lower case letters used above indicate scalars which are 1 x 1 matrices. Now in the case that « > 1 , we can no longer be sure that P(t)=K a constant matrix, but we can obtain a result which in the scalar case imlies p(t)=k. To see that P(t)=Kmay fail when (1) and (2) are true, we consider the following example. Let