If true, the following conjecture of Thomassen [Th81] is a planarity criterion for a special class of graphs that involves only K 5. Recall that a planar graph on n vertices contains at most 3n-6 ...

Let G be a connected graph that is 2-cell embedded in some surface. For a vertex v and a face f of G, let deg(v) denote the degree of v and let deg(f) denote the size of f. We define the combinatorial ...

We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected ...