Techniques for the refinement of orthogonal graph drawings.
The absence of efficient dual pairs of spanning trees in planar graphs.
The Analysis of a Nested Dissection Algorithm.
The block plan problem. A graphtheoretic approach
The Chromatic Number of a Class of Pseudo-2-Manifolds.
The crossing number of a projective graph is quadratic in the face-width.
The crossing number of the generalized Petersen graph is four
The crossing number of the generalized Petersen graph
Guy and Harary (1967) have shown that, for , the graph is homeomorphic to the Möbius ladder , so that its crossing number is one; it is well known that is planar. Exoo, Harary and Kabell (1981) have shown hat the crossing number of is three, for Fiorini (1986) and Richter and Salazar (2002) have shown that has crossing number two and that has crossing number , provided . We extend this result by showing that also has crossing number for all .
The crossing numbers of certain Cartesian products
In this article we determine the crossing numbers of the Cartesian products of given three graphs on five vertices with paths.
The crossing numbers of join products of paths with graphs of order four
Kulli and Muddebihal [V.R. Kulli, M.H. Muddebihal, Characterization of join graphs with crossing number zero, Far East J. Appl. Math. 5 (2001) 87-97] gave the characterization of all pairs of graphs which join product is planar graph. The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. There are only few results concerning crossing numbers of graphs obtained as join product of two graphs. In the paper, the exact values of crossing numbers...
The crossing numbers of products of a 5-vertex graph with paths and cycles
There are several known exact results on the crossing numbers of Cartesian products of paths, cycles or stars with "small" graphs. Let H be the 5-vertex graph defined from K₅ by removing three edges incident with a common vertex. In this paper, we extend the earlier results to the Cartesian products of H × Pₙ and H × Cₙ, showing that in the general case the corresponding crossing numbers are 3n-1, and 3n for even n or 3n+1 if n is odd.
The Crossing Numbers of Products of Path with Graphs of Order Six
The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. For the path Pn of length n, the crossing numbers of Cartesian products G⃞Pn for all connected graphs G on five vertices are also known. In this paper, the crossing numbers of Cartesian products G⃞Pn for graphs G of order six are studied. Let H denote the unique tree of order six with two vertices of degree three. The main contribution is that the crossing number of the Cartesian...
The crossing numbers of some generalized Petersen graphs.
The decay number and the maximum genus of a graph
The edge-minimal polyhedral maps of Euler characteristic .
The evolution of uniform random planar graphs.
The face labeling of maximal planar graphs.
The fundamental group and Galois coverings of hexagonal systems in 3-space.
The fundamental group of a locally finite graph with ends-a hyperfinite approach
The end compactification |Γ| of a locally finite graph Γis the union of the graph and its ends, endowed with a suitable topology. We show that π₁(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct factors through an embedding into an inverse limit of free groups. We also show how to recover the standard description of π₁(|Γ|) given by Diestel and Sprüssel (2011). Finally, we give...
The Gauss code problem off the plane.