Showing papers on "Quartic graph published in 1972"
TL;DR: An algorithm to enumerate all the elementary circuits of a directed graph that uses back-tracking with lookahead to avoid unnecessary work and has a time bound of $O ((V+E)(C+1))$ when applied to a graph with V, vertices, edges, and elementary circuits.
Abstract: An algorithm to enumerate all the elementary circuits of a directed graph is presented. The algorithm uses back-tracking with lookahead to avoid unnecessary work, and it has a time bound of $O ((V+E)(C+1))$ when applied to a graph with $V$ vertices, $E$ edges, and $C$ elementary circuits. Keywords: Algorithm, circuit, cycle, graph
278 citations
01 Jan 1972
TL;DR: In this paper, the non-planar planner maps are defined as a set of edges such that the rest of the graph consists of two disjoint trees, and each edge of H has one end in each tree.
Abstract: Publisher Summary This chapter discusses the non-Hamiltonian planner maps. A planar map is a dissection of the sphere or closed plane into a finite number of simply connected polygonal regions called faces or countries by means of a graph drawn in the surface. It is assumed that this graph has no loop or isthmus. A Hamiltonian circuit in a map is a circuit in its graph passing through every vertex. A map is called Hamiltonian or non-Hamiltonian according to as it does or does not have such a circuit. A Hamiltonian bond in a graph G is a set H of edges such that the rest of the graph consists of two disjoint trees, and each edge of H has one end in each tree. Denoting the number of vertices of G of valency i by f i , and suppose f i ' of these to be in the first tree and f i ’ in the second. This form of the theory applies to all graphs, whether planar or nonplanar.
10 citations
TL;DR: In this paper, the density of a given family of subsets of an abstract set S in another set S is deduced for the same set in a normed linear function space, and then the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces are applied to functional on Hilbert spaces.
Abstract: In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of \"triangle functional \" (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications. Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].
9 citations
TL;DR: In this article, an inductive definition of the property that a graph which cannot be colored with n colors is given, where n is the number of colors in the input graph.
Abstract: We give an inductive definition of the property “a graph which cannot be colored with n colors.”
3 citations
TL;DR: In this paper, it was shown that if the graph G of a network N is the sum of n Hamiltonian circuits, the product G×G of G with itself is a 2n Hamiltonian circuit.
Abstract: It is shown that, if the graph G of a network N is the sum of n Hamiltonian circuits, the product G×G of G with itself is the sum of 2n Hamiltonian circuits For example, C×C, where C is a circuit, is the sum of two Hamiltonian circuits, and K2n+1×K2n+1, where K2n+1 is the complete graph of order 2n+1, is the sum of 2n such circuits
TL;DR: By virtue of these definitions the authors obtain a necessary and sufficient condition for that an oriented graph G(X, Γ) of even order has a perfect matching and similarly for that a Hamiltonian circuit.
Abstract: Some fundamental properties of a graph are defined in terms of the edge-edge incidence matrix associated with the graph. By virtue of these definitions we obtain a necessary and sufficient condition for that an oriented graph G(X, Γ) of even order has a perfect matching and similarly for that a Hamiltonian circuit. Not only for a simple graph but also for any oriented graph of even order, each of these necessary and sufficient condition is useful for knowing the existence of a perfect matching and a Hamiltonian circuit, respectively.