Catacondensed hexagonal systems with smaller numbers of Kekule structures

In this paper, we investigate the catacondensed hexagonal systems with smaller numbers of Kekule structures. The unbranched catacondensed hexagonal systems with one kink are ordered. The unbranched catacondensed hexagonal system with two kinks and the branched catacondensed hexagonal system with one branched hexagon, which have the smallest and largest numbers of Kekule structures, are determined. Furthermore, based on the above results, the catacondensed hexagonal systems with the first up to the fourth smallest numbers of Kekule structures are determined. (C) 2003 Elsevier B.V. All rights reserved

Rankings of graphs

A vertex (edge) coloring c : V ! f1; 2; : : : ; tg (c 0 : E ! f1; 2; : : : ; tg) of a graph G = (V; E) is a vertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number Ø r (G) (edge ranking number Ø 0 r (G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the vertex ranking and edge ranking problems. Among others it is shown that Ø r (G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k. We characterize those graphs where the vertex ranking number Ø r and the chromatic number Ø coincide on all induced subgraphs, show that Ø r (G) = Ø(G) implies Ø(G) = !(G) (largest clique size) and give a formula for Ø 0 r (Kn )

Rankings of graphs

A vertex (edge) coloring c: V!f1; 2;:::;tg (c 0: E!f1; 2;:::; tg) of a graph G =(V;E) isavertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number r(G) (edge ranking number 0 r(G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexityofthevertex ranking and edge ranking problems. Among others it is shown that r(G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any xed k. We characterize those graphs where the vertex ranking number r and the chromatic number coincide on all induced subgraphs, show that r(G) = (G) implies (G) =!(G) (largest clique size) and give a formula for 0 r(K n)

On-line Algorithms for a Single Machine Scheduling Problem

An increasingly significant branch of computer science is the study of online algorithms. In this paper, we apply the theory of on-line algorithms to job scheduling. In particular, we study the nonpreemptive single machine scheduling of independent jobs with arbitrary release dates to minimize the total completion time. We design and analyze two on-line algorithms which make scheduling decisions without knowing about jobs that will arrive in future. Keywords: job scheduling, on-line algorithm, c-competitiveness 1 Introduction Given a sequence of requests, an on-line algorithm is one that responds to each request in the order it appears in the sequence without the knowledge of any request following it in the sequence. For instance, in the bin packing problem, a list L = (a 1 ; a 2 ; : : : ; a n ) of reals in (0; 1] needs to be packed into the minimum number of unit-capacity bins. An on-line bin packing algorithm packs a i , where i starts from 1, without knowing about a i+1 ; : : : ; ..