Showing papers on "Quartic graph published in 1975"

Finding All the Elementary Circuits of a Directed Graph

Abstract: An algorithm is presented which finds all the elementary circuits of a directed graph in time bounded by $O((n + e)(c + 1))$ and space bounded by $O(n + e)$, where there are n vertices, e edges and c elementary circuits in the graph. The algorithm resembles algorithms by Tiernan and Tarjan, but is faster because it considers each edge at most twice between any one circuit and the next in the output sequence.

Advances on the Hamiltonian Completion Problem

Abstract: The Hamiltonian completion problem for an arbitrary graph G consists of determining the minimum number of new lines which can be added to G in order to produce a Hamiltonian cycle in G. A solution to this problem would be useful in situations where it is necessary to periodically traverse a network or data structure in such a way as to visit all nodes and minimize the length of the traversal. Linear algorithms are presented for solving the Hamiltonian completion problem for several classes of graphs, in particular for trees and unicyclic graphs. Several more general results are also given.

On the number of Hamiltonian circuits in the $n$-cube

Abstract: Improved upper and lower bounds are found for the number of hamiltonian circuits in the n-cube.

Applications of graph theory to matrix theory

Abstract: Let A l Ak be n x n matrices over a commutative ring R with identity. Graph theoretic methods are established to compute the standard polynomial [A , ... I Ak]. It is proved that if k < 2n 2, and if the characteristic of R either is zero or does not divide 4I(V2 n) 2, where I denotes the greatest integer function, then there exist n x n skew-symmetric matrices A 1 . . . , Ak such that [A 1, . . . AkI AO.

The Entire Graph of a Bridgeless Connected Plane Graph is Panconnected

Abstract: Recently A. M. Hobbs and J. Mitchem [7] proved that the entire graph of a bridge-less connected plane graph is Hamiltonian. In this paper we strengthen this result substantially by showing that entire graphs of such plane graphs are panconnected. (Between each pair of distinct vertices in a panconnected graph there exist paths of all lengths greater than or equal to the distance between the vertices.) This fits a pattern which indicates that Hamiltonian-connected graphs seem to have paths of \" many \" lengths between each pair of distinct points [1,2, 3]. The graphs we consider will be undirected, finite, and have no loops or multiple edges. A plane graph is a graph already embedded in the plane. If G is a plane graph, V(G), E(G) and F(G) denote the sets of its vertices, edges and faces, respectively. Two distinct vertices (edges, faces) of G are adjacent if they share a common edge (vertex, edge). A vertex and an edge, a vertex and a face, or an edge and a face, are adjacent if they are incident (in the obvious sense). The entire graph of G, denoted e(G), is the graph with vertex set V(G) u E(G) u F(G), with two vertices of e(G) adjacent if and only if they are adjacent in G. Hamiltonian and Eulerian properties of entire graphs were first discussed by J. Mitchem in [8]. By P(l) (respectively C(/)) we will mean a path (respectively circuit, i.e., cycle) with /vertices. A path with (distinct) vertices v u v 2 ,..., v t and edges e u e 2 ,..., e x-\\ will be written (y l5 v 2 ,..., v t) or [v lt e lt v 2> e 2 ,..., e,_ l5 v t ], while (v it v 2> ..., v t , Vi) denotes a circuit with the same vertices. If P = (v 2 , v 3 ,..., u,), then (v u P, v l+ x) denotes the path (v l9 v 2 ,..., v h v l+l) (if it exists). If u ^ v, d G (u, v) will denote the distance between u and v in G, while P,(w, V) will denote a path between u and v containing / vertices. If P,(w, v) exists for all u ^ v in V(G) and for all /, d G (u, v) < I ^ | V(G)\\, then G is called panconnected. …