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.
The genus of nearly complete graphs-case 6.
The intersection numbers of a complex.
The kernel operation on subsets of a -space
The L²-invariants and Morse numbers
We study the homotopy invariants of free cochain complexes and Hilbert complexes. These invariants are applied to calculation of exact values of Morse numbers of smooth manifolds.
The list linear arboricity of planar graphs
The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having Δ ≥ 13, or for any planar graph with Δ ≥ 7 and without i-cycles for some i ∈ {3,4,5}. We also prove that ⌈½Δ(G)⌉ ≤ lla(G) ≤ ⌈½(Δ(G)+1)⌉ for any planar graph having Δ ≥ 9.
The map asymptotics constant .
The maximum genus, matchings and the cycle space of a graph
In this paper we determine the maximum genus of a graph by using the matching number of the intersection graph of a basis of its cycle space. Our result is a common generalization of a theorem of Glukhov and a theorem of Nebeský .
The maximum of the maximum rectilinear crossing numbers of -regular graphs of order .
The minor crossing number of graphs with an excluded minor.
The neighborhood complex of an infinite graph.
The non-crossing graph.
The number of archiral trees.
The number of crossings in a regular drawing of the complete bipartite graph.
The Number of Labeled Dissections of a k-Ball.
The planarity theorems of MacLane and Whitney for graph-like continua.