The circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant, harmonic, bounded, Dirichlet function. If P is a bounded circle packing in R 2 whose contacts graph is a bounded valence triangulation of a disk, then, with probability 1, the simple random walk on P converges to a limit point. Moreover, in this situation any continuous fun...

A nonplanar graph G is near-planar if it contains an edge e such that G− e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers an...

Given two graphs G1 and G2, the planar Ramsey number PR(G1, G2) is the smallest integer n such that every planar graph on n vertices either contains a copy of G1 or its complement contains a copy of G2. So far, the planar Ramsey numbers have been determined, when both, G1 and G2 are complete graphs or both are cycles. By combining computer search with some theoretical results, in this paper we ...

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. In general, 1-planar graphs do not admit straightline drawings. We show that every 3-connected 1-planar graph has a straight-line drawing on an integer grid of quadratic size, with the exception of a single edge on the outer face that has one bend. The drawing can be computed in linear time from any...

We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning ...

We prove that the order of canonical vector bundle over configuration space is $2$ for a general planar graph, and $4$ nonplanar graph.

We present a new algorithm to test whether a given graph G is planar and to compute a planar embedding Ĝ of G if such an embedding exists. Our algorithm utilizes a fundamentally new approach based on graph separators to obtain such an embedding. The I/O-complexity of our algorithm is O(sort(N)). A simple simulation technique reduces the I/O-complexity of our algorithm to O(perm(N)). We prove a ...

In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is NP-hard, even for planar graphs of maximum degree 4, and we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Further, we show that every planar graph has a subdivision with one vertex per edge that admits an acyclic 3-colori...

We investigate the computational complexity of the following problem. Given a “mixed” planar graph, i.e., a planar graph where some edges are directed, orient the remaining edges in such a way that the whole graph is acyclic and admits a bimodal planar embedding, that is, a planar embedding where edges entering (exiting) each vertex appear consecutively in its circular adjacency list. We show t...

A dynamic chromatic number χd(G) of a graph G is the least number k such that G has a proper k-coloring of the vertex set V (G) so that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. We show that χd(G) ≤ 4 for every planar graph except C5, which was conjectured in [5]. The list dynamic chromatic number chd(G) of G is the least number k such that for an...