planar graph

Bounded colorings for planar graphs ★★

Author(s): Alon; Ding; Oporowski; Vertigan

Question   Does there exists a fixed function $ f : {\mathbb N} \rightarrow {\mathbb N} $ so that every planar graph of maximum degree $ d $ has a partition of its vertex set into at most three sets $ \{V_1,V_2,V_3\} $ so that for $ i=1,2,3 $, every component of the graph induced by $ V_i $ has size at most $ f(d) $?

Keywords: coloring; partition; planar graph

Universal point sets for planar graphs ★★★

Author(s): Mohar

We say that a set $ P \subseteq {\mathbb R}^2 $ is $ n $-universal if every $ n $ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $ P $, and all edges are (non-intersecting) straight line segments.

Question   Does there exist an $ n $-universal set of size $ O(n) $?

Keywords: geometric graph; planar graph; universal set

What is the largest graph of positive curvature?

Author(s): DeVos; Mohar

Problem   What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or antiprism?

Keywords: curvature; planar graph

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