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1. Some background knowledge Definition. A torus, informally, is the doughnut-shaped surface that we get by taking a square made out of some arbitrarily-stretchy material and gluing together opposite …

A 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. 1 planar graph I read the f …

A graph is $k$-planar if it can be drawn on the plane such that each edge is crossed at most $k$ times. A graph together with a $k$-planar drawing is a $k$-plane graph. Hence, by definition, $0$-plana …

A graph is 1-planar if it can be drawn on the plane such that each edge is crossed at most once. Let $G$ be a 1-planar bipartite graph with $n~(n > 4)$ vertices and $m$ edges. Karpov [1] showed that $ …

It is well known that every maximal planar graph with at least 4 vertices is 3-connected. But for maximal 1-planar graphs we cannot ensure the high connectivity. (See is-there-any-maximal-1-planar-or …

The line graph of an undirected graph $G$ is another graph $L(G)$ that represents the adjacencies between edges of $G$. $L(G)$ is constructed in the following way: for each edge in $G$, make a vertex …

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. A graph $G$ is maximal in a graph clas …

In graph drawing and geometric graph theory, a Tutte embedding of a simple 3-vertex-connected planar graph is a crossing-free straight-line embedding with the properties that the outer face is a conve …

Sometimes I look at all non-isomorphic good drawings of graphs on a plane or sphere. Good drawing means that no edge crosses itself, no two edges cross more than once, and no two edges incident with t …

Thanks for advice from Timothy Chow. I have now received an email from CRWS. The graph has a crossing number of 12. Its crossing-minimal drawing is as follows.

From the Wikipedia entry on Tutte 12-cage , it is stated that the crossing number of Tutte 12-cage is 170, but the cited references do not seem to provide sufficient explanation for this. Exoo, G. " …

A $1$-planar graph is a graph which has a drawing on the plane such that each edge has at most one crossing. I used nauty to generate all 3-regular graphs up to order 12, and checked each one of them …

In the following paper, Hudák Dávid, and Tomáš Madaras give the following Theorem 1.1. Hudák, Dávid, and Tomáš Madaras. "On local properties of 1-planar graphs with high minimum degree." Ars Mathemat …

A $k$-planar graph is a graph which can be embedded with at most $k$ crossings per edge. It is proved that a complete graph $K_n$ is 2-planar if and only if $n\le 7$. Angelini P., Bekos M. A., Kaufma …

1-planar graphs were first studied by Ringel (1965), who showed that they can be colored with at most seven colors. Later, the precise number of colors needed to color these graphs, in the worst case, …