On the Maximum Degree of Bipartite Embeddings of Trees in the Plane

Abstract

A geometric graph G =(V(G),E(G)) is a graph drawn in the plane such that V(G) is a set of points in the plane, no three of which are tollinear, and E(G) is a set of (possibly crossing) straight-line segments whose endpoints belang to V(G). If a geometric graph G is a complete bipartite graph with partite sets A and B, i.e., V(G) = A ∪ B, then G is denoted by K(A, B). Let A and B be two disjoint sets of points in the plane such that |A| = |B| and no three points of A ∪ B are tollinear. Then we show that the geometric complete bipartite graph K(A, B) contains a spanning tree T without crossings such that the maximum degree of T is at most 3.