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A planar graph is one which has a plane embedding. Two drawings are topologically isomorphic if one can be continuously deformed into the other. If we wrap a drawing onto a sphere, and then off again, …

Thanks for Brendan McKay's help. The following two are indeed not same embedding. When the subgraph on the right is reversed to the 3 face of the graph on the left subgraph.

When I looked up the definition of convex drawing of planar graph, my confusion mainly focused on the outer face. The following definition of convex drawing is from Wikipedia. In graph drawing, a co …

The plantri (see http://users.cecs.anu.edu.au/~bdm/plantri/) is a program that generates certain types of graphs that are imbedded on the sphere. Exactly one member of each isomorphism class is output …

My question may be similar to generating-21-vertex-4-regular-plane-graphs-with-8-faces-of-degree-3-and-15-face., but it has differences. The plane graphs I desire (without needing regularity) have onl …

I asked this question on math stack, but didn't get any response, so I ask it here. In a previous post, I proved that no 5-connected maximal planar graph is perfect. (A perfect graph is a graph $G$ su …

In 1979, Hakimi and Schmeichel [1] initiated such a study by determining the maximum number of triangles and 4-cycles possible in an $n$-vertex planar graph (see also [2] for a small correction). [1] …

The well-known max cut problem asks for a largest cut of a graph $G$. A cut of maximal size clearly corresponds to a bipartite subgraph of maximal size. After my inquiry, in planar graphs, the maximum …

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 …

My question was partly inspired by the question linked below. There is a 3-connected 5-regular simple $n$-vertex planar graph iff $n$ satisfies....? I see a wonderful construction of Adam P. Goucher …

The subsequent statements are extracted from the article titled 'Generating r-regular graphs' (https://doi.org/10.1016/S0166-218X(02)00593-0). A well-known classical theorem of Steinitz and Rademache …

Tutte proved the famous result: Every planar 4-connected graph has a hamiltonian cycle. But I read in Section 111.6.5 on book Eulerian Graphs and Related Topics that the author Herbert Fleischner pl …

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 Halin graph is a graph constructed by embedding a tree with no vertex of degree two in the plane and then adding a cycle to join the tree’s leaves. We found a list of the number of Halin graphs with …