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Graphs with Large Disjunctive Total Domination Number
Let be a graph with no isolated vertex. In this paper, we study a
parameter that is a relaxation of arguably the most important domination
parameter, namely the total domination number, . A set of
vertices in is a disjunctive total dominating set of if every vertex is
adjacent to a vertex of or has at least two vertices in at distance
from it. The disjunctive total domination number, , is the
minimum cardinality of such a set. We observe that . Let be a connected graph on vertices with minimum degree
. It is known [J. Graph Theory 35 (2000), 21--45] that if and , then . Further [J. Graph Theory 46
(2004), 207--210] if , then . We prove that
if and , then and we
characterize the extremal graphs.Comment: 50 page
Scalar curvature estimates for compact symmetric spaces
We establish extremality of Riemannian metrics g with non-negative curvature
operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let
g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors.
We show that k'\ge k everywhere on M implies k'=k. Under an additional
condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study
area-non-increasing spin maps onto such Riemannian manifolds.Comment: 13 pages, LaTeX, uses amsar
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