Author
Stephen C. Locke
Other affiliations: University of Waterloo
Bio: Stephen C. Locke is an academic researcher from Florida Atlantic University. The author has contributed to research in topics: Path graph & k-vertex-connected graph. The author has an hindex of 12, co-authored 36 publications receiving 498 citations. Previous affiliations of Stephen C. Locke include University of Waterloo.
Papers
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TL;DR: In this article, an algorithm polynomial for determining a bipartite sous-graphe bipartitio d'un graphe G without triangle and a Boucle de degre maximum 3, contenant au moins 4/5 des aretes de G.
Abstract: On presente un algorithme polynomial permettant de determiner un sous-graphe biparti d'un graphe G sans triangle ni boucle de degre maximum 3, contenant au moins 4/5 des aretes de G. On caracterise le dodecaedre et le graphe de Petersen comme les seuls graphes connexes 3-reguliers sans triangle ni boucle pour lesquels il n'existe pas de sous graphe biparti ayant un nombre d'aretes superieur a cette proportion
86 citations
TL;DR: The subchromatic number X s (G ) ⩽ (G ), defined in this article, is the smallest order k of a partition of a graph such that the subgraph (V i ) induced by each subset V i consists of a disjoint union of complete subgraphs.
66 citations
TL;DR: A lower bound is established on the number of edges in a maximum k-colorable subgraph of a loopless graph G that is a bipartite subgraph whose color classes are of equal size.
Abstract: A lower bound is established on the number of edges in a maximum k-colorable subgraph of a loopless graph G. For the special case of 3-regular graphs, lower bounds are also determined on the maximum number of edges in a bipartite subgraph whose color classes are of equal size.
39 citations
TL;DR: Let G be a k-connected graph with minimum degree d and at least 2d vertices and G has a cycle of length at least 1d through any specified set of k vertices.
36 citations
TL;DR: The Hamilton space of any connected Cayley graph on an abelian group is determined in this paper and is shown to be a subspace of the cycle space called the Hamilton space.
33 citations
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Book•
17 Dec 1994
TL;DR: In this article, the Conjectures of Hadwiger and Hajos are used to define graph types, such as planar graph, graph on higher surfaces, and critical graph.
Abstract: Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.
1,380 citations
TL;DR: The second edition of the ONAG book as mentioned in this paper presents recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.
Abstract: ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.
605 citations
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TL;DR: This work presents data which, to the best of its knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge.
Abstract: We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values.
581 citations
TL;DR: In this paper, Nordhaus and Gaddum gave lower and upper bounds on the sum and product of the chromatic number of a graph and its complement, in terms of the order of the graph.
198 citations
TL;DR: In this paper, the complexity of finding a long path in directed or undirected graphs was studied and an algorithm for finding a longest path with complexity O(K! · |V| · |E|) was proposed.
Abstract: We study the complexity of finding long paths in directed or undirected graphs, Given a graph G =(V, E) and a number k our algorithm decides within time O(K! · |V| · |E|) for all u,v ɛ V Whether there exists some path of length k firm u to v. The complexity of this algorithm has to be compared with 0(|V| k−1 · |E|) Which is the worst case behaviour of the algorithms described up to now in the literature, We get similar results for the problems of finding a longest path, a cycle of length k or a longest cycle, respectively. Our approach is based on the idea of representing certain families of sets by subfamilies of small cardiality. We also discuss the border lines of this idea.
179 citations