Showing papers on "Connectivity published in 1982"
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TL;DR: It is shown that there is a subsetd ⊂V Γ which has the following properties: bothd andd*=VΓ\d are infinite.
Abstract: LetΓ be infinite connected graph with more than one end. It is shown that there is a subsetd ⊂V Γ which has the following properties. (i) Bothd andd*=VΓ\d are infinite. (ii) there are only finitely many edges joiningd andd*. (iii) For eachge AutΓ at least one ofd⊂dg, d*⊂dg, d⊂d* g, d*⊂d* g holds. Any group acting on Γ has a decomposition as a free product with amalgamation or as an HNN-group.
90 citations
TL;DR: The method of Lee and the simple application of graph theory seem to give different answers for flow networks as mentioned in this paper, and the method whereby graph theory can be extended to give correct results is explained.
Abstract: The method of Lee and the simple application of graph theory seem to give different answers for flow networks. Lee's method is correct. The method whereby graph theory can be extended to give correct results is explained.
25 citations
TL;DR: In this article, it was shown that if Tn is replaced by any sparse connected graph G on n vertices and K3 is replaced with an odd cycle Ck, then for appropriate n the Ramsey number is unchanged.
Abstract: Introduction. Let G and H be simple graphs. The Ramsey number r(G, H) is the smallest integer n such that for each graph F on n vertices, either G is a subgraph of F or H is a subgraph of F, the complement of F. Calculation of r(G, H) for particular pairs of graphs G and H has received considerable attention, and a survey of such results can be found in [2]. Chvatal [5] proved that if Tn is a tree on n vertices and Km is a complete graph on m vertices, then r(T7,, Kin) = (n 1)(m 1) + 1. In [4] it was shown that if Tn is replaced by a sparse connected graph Gn on n vertices the Ramsey number remains the same (i.e. r(G,, Kin) = (n 1)(m 1) + 1). For m = 3 Chvatal's theorem implies r(T1, K3) = 2n -1. In this paper we will show that if Tn is replaced by any sparse connected graph G on n vertices and K3 is replaced by an odd cycle Ck, then for appropriate n the Ramsey number is unchanged. In particular we will prove the following.
23 citations
TL;DR: In this article, the authors consider only finite, undirected graphs without loops or multiple edges and define a subspace of the graph called the cycle space of G, denoted by Q(G).
15 citations
TL;DR: A nonisomorphic, edge-hypomorphic pair of countable forests is constructed, providing a counterexample to the edge-reconstruction conjecture for infinite graphs that is simpler than the countereXamples previously given by C. Thomassen.
5 citations
TL;DR: It is proved that if @C is a finite connected graph having the same convex subgraphs as the graph of the n-dimensional cube Q"n (n>=3), then |V(@C)|>=|V(Q"n)|.
4 citations
TL;DR: In this article, a two-person game related to the connectivity among all vertices of a graph is defined, and necessary and sufficient conditions for the short, cut and neutral games are given in terms of the principal partition of the graph.
Abstract: A two-person game related to the connectivity among all vertices of a graph is defined. Necessary and sufficient conditions for the short, cut and neutral games are given in terms of the principal partition of a graph. Winning strategies are described.
3 citations
TL;DR: It is proved that if n is large enough, the Ramsey number r(G"1,...,G"c,H"n) has the form (X-1)(n-1)+T, where X and T are two Ramsey-type functons involving G"1, G"c only.
3 citations
TL;DR: The three types of primitive sets in a connected graph are compared and it is proved that each complete lattice can be represented as an H -lattice and each P-Iattice as a W - lattice.
2 citations
01 Jan 1982
TL;DR: In this article, the derivation of high temperature series expansions for the Ising model, with special reference to the spin ½ case, is discussed, and the methods of derivation can be extended to the classical vector model.
Abstract: In this talk, I shall discuss the derivation of high temperature series expansions for the Ising model, with special reference to the spin ½ case. In the second lecture, we shall see how these methods of derivation can be extended to the classical vector model.
1 citations