Results 1 
4 of
4
The Traveling Salesman Problem for Cubic Graphs
, 2004
"... We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2 n/3) ≈ 1.260 n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycl ..."
Abstract

Cited by 27 (2 self)
 Add to MetaCart
We show how to find a Hamiltonian cycle in a graph of degree at most three with n vertices, in time O(2 n/3) ≈ 1.260 n and linear space. Our algorithm can find the minimum weight Hamiltonian cycle (traveling salesman problem), in the same time bound. We can also count or list all Hamiltonian cycles in a degree three graph in time O(2 3n/8) ≈ 1.297 n. We also solve the traveling salesman problem in graphs of degree at most four, by randomized and deterministic algorithms with runtime O((27/4) n/3) ≈ 1.890 n and O((27/4+ǫ) n/3) respectively. Our algorithms allow the input to specify a set of forced edges which must be part of any generated cycle. Our cycle listing algorithm shows that every degree three graph has O(2 3n/8) Hamiltonian cycles; we also exhibit a family of graphs with 2 n/3 Hamiltonian cycles per graph.
A Constructive Enumeration of Fullerenes
"... In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regul ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
In this paper, a fast and complete method to enumerate fullerene structures is given. It is based on a topdown approach, and it is fast enough to generate, for example, all 1812 isomers of C 60 in less than 20 seconds on an SGIworkstation. The method described can easily be generalised for 3regular spherical maps with no face having more than 6 edges in its boundary.
A Conjecture of Erdös the Ramsey Number r(W6)
 J. Combinatorial Math. and Combinatorial Computing
, 1996
"... It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by ve ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17; where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels. 1 Introduction The following well known conjecture is due to Paul Erdos. CONJECTURE 1 If G is a graph with chromatic number Ø(G) k; then the Ramsey number r(G) r(K k ): The strong form of the Erdos conjecture is that if Ø(G) k; and G does not contain a copy of K k ; then r(G) ? r(K k ). For k = 3 it is trivial to verify this stronger conjecture. If G 6' K 3 and Ø(G) 3; then G has at least 4 vertices. Thus r(G) ? 6 = r(K 3 ); since neither the graph K 3 [K...
The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...