Ramsey numbers of graphs with long tails

AbstractThe ramsey number of a connected nonbipartite graph G with a sufficiently long path emanating from one of its points is found to be (n−1)(χ−1)+s, where n is the number of points of G, χ is the chromatic number of G, and s is the minimum possible number of points in a color class in a χ-coloring of the points of G

Generalized ramsey theory for graphs, x: double stars

The double star S(n, m), where n [ges] m [ges] 0, is the graph consisting of the union of two stars K1,n and K1,m together with a line joining their centers. Its ramsey number r(S(n, m)) is the least number p such that there is a monochromatic copy of S(n, m) in any 2-coloring of the edges of Kp. It is shown that r(S(n, m)) = max (2n + 1, n + 2m + 2) if n is odd and m[les]2 and r(S(n, m)) = max (2n + 2, n + 2m + 2) otherwise, for n [les] [radical sign]2m or n [ges] 3m.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23705/1/0000677.pd

The Economic Consequences of Social-Network Structure

We survey the literature on the economic consequences of the structure of social networks. We develop a taxonomy of "macro" and "micro" characteristics of social-interaction networks and discuss both the theoretical and empirical findings concerning the role of those characteristics in determining learning, diffusion, decisions, and resulting behaviors. We also discuss the challenges of accounting for the endogeneity of networks in assessing the relationship between the patterns of interactions and behaviors

Dominating Sets Whose Closed Stars Form Spanning Trees

For a subset W of vertices of an undirected graph G, let S(W ) be the subgraph consisting of W , all edges incident to at least one vertex in W , and all vertices adjacent to at least one vertex in W . If there exists a W such that S(W ) is a tree containing all the vertices of G, then S(W ) is a spanning star tree of G. These and associated notions are related to connected and/or acyclic dominating sets and also arise in the study of A-trails in Eulerian plane graphs. Among the results in this paper are a characterization of those values of n and m for which there exists a connected graph with n vertices and m edges that has no spanning star tree, and a proof that finding spanning star trees is in general NP-hard. AMS Subject Classification (1991): Primary: 05C35 Secondary: 05C05, 05C45, 05C85, 68R10, 90B12 1. Introduction In this paper we introduce a new variation on domination in graphs. The motivation for this research grew not from the wealth of results on dominating sets---s..

Spanning Star Trees In Regular Graphs

For a subset W of vertices of an undirected graph G, let S(W ) be the subgraph consisting of W , all edges incident to at least one vertex in W , and all vertices adjacent to at least one vertex in W . If S(W ) is a tree containing all the vertices of G, then we call it a spanning star tree of G. In this case W forms a weakly connected but strongly acyclic dominating set for G. We prove that for every r # 3, there exist r-regular n-vertex graphs that have spanning star trees, and there exist r-regular n-vertex graphs that do not have spanning star trees, for all n su#ciently large (in terms of r). Furthermore, the problem of determining whether a given regular graph has a spanning star tree is NP-complete. AMS Subject Classification (1991): Primary: 05C35 Secondary: 05C05, 05C85 Key words: dominating set, weakly connected, strongly acyclic, regular graph 1. Introduction In this paper we continue the study of weakly connected, strongly acyclic domination in graphs, introduced i..

The Evolution of the Mathematical Research Collaboration Graph

We discuss some properties of the research collaboration graph for mathematicians, look at its evolution over time, and survey some random models that might produce graphs of this sort. Our approach is more experimental and statistical than theoretical. Further information is available on the Erdős Number Project web site (especially the subpag

Ackermann Function, or What's So Special about 1969?

this article can provide a more definitive description of what's going on. To be specific, let N denote the se