The notion of 1-planarity is among the most natural and most studied generalizations of
graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at …

After a brief historical account, a few simple structural theorems about plane graphs useful
for coloring are stated, and two simple applications of discharging are given. Afterwards, the …

A graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by
at most one other edge. In the paper, we study the existence of subgraphs of bounded …

In 1965 Ringel raised a 6 color problem for graphs that can be stated in at least three
different forms. In particular, is it possible to color the vertices and faces of every plane graph …

A graph is 1‐planar if it can be drawn on the plane so that each edge is crossed by no more
than one other edge (and any pair of crossing edges cross only once). A non‐1‐planar …

It is well known that every planar graph contains a vertex of degree at most 5. A theorem of
Kotzig states that every 3-connected planar graph contains an edge whose endvertices …

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions
of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we …

We give new results about the relationship between 1-planar graphs and RAC graphs. A
graph is 1-planar if it has a drawing where each edge is crossed at most once. A graph is …

Graphs can be represented in many di erent ways: by lists of edges, by incidence relations,
by adjacency matrices, and by other similar structures. These representations are well suited …

We consider a modified notion of planarity, in which two nations of a map are considered
adjacent when they share any point of their boundaries (not necessarily an edge, as …