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A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . . , k so that, for each face α of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.
Menger's Theorem for digraphs states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t + 1 disjoint A-B-paths in D. Here a short and elementary proof of a more general theorem is given. For terminology and notation not deened here we refer to 1]. Recall that for a directed… (More)
Given a connected graph G = (N, E) with node weights s ∈ RN+ and nonnegative edge lengths, we study the following embedding problem related to an eigenvalue optimization problem over the second smallest eigenvalue of the (scaled) Laplacian of G: Find vi ∈ R , i ∈ N so that distances between adjacent nodes do not exceed prescribed edge lengths, the weighted… (More)
We study the problem of maximizing the second smallest eigenvalue of the Laplace matrix of a graph over all nonnegative edge weightings with bounded total weight. The optimal value is the absolute algebraic connectivity introduced by Fiedler, who proved tight connections of this value to the connectivity of the graph. Using semidefinite programming… (More)
Let F be a set of graphs and for a graph G let αF(G) and α∗ F(G) denote the maximum order of an induced subgraph of G which does not contain a graph in F as a subgraph and which does not contain a graph in F as an induced subgraph, respectively. Lower bounds on αF(G) and α∗ F(G) and algorithms realizing them are presented.
A short proof of the classical theorem of Menger concerning the number of disjoint AB-paths of a nite digraph for two subsets A and B of its vertex set is given. c © 2000 Elsevier Science B.V. All rights reserved. MSC: 05C40
For an integer d ≥ 3 let α(d) be the supremum over all α with the property that for every > 0 there exists some g( ) such that every d-regular graph of order n and girth at least g( ) has an independent set of cardinality at least (α− )n. Extending an approach proposed by Lauer and Wormald (Large independent sets in regular graphs of large girth, J. Comb.… (More)
The well-known lower bound on the independence number of a graph due to Caro (New Results on the Independence Number, Technical Report, TelAviv University, 1979) and Wei (A Lower Bound on the Stability Number of a Simple Graph, Technical memorandum, TM 81 11217 9, Bell laboratories, 1981) can be established as a performance guarantee of two natural and… (More)
In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance… (More)
For c 2 and k minfc; 3g, guaranteed upper bounds on the length of a shortest cycle through k prescribed vertices of a c-connected graph are proved. Analogous results on planar graphs are presented, too.