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 collinear, and E(G) is a set of (possibly crossing) straight-line segments whose endpoints belong to V(G). If a geometric graph G is a complete bipartite graph with partite sets A and B, i.e., V(G) = AuB, then G is denoted by K(A, B). Let A and B be two disjoint sets of points in the plane such that |^4| = |jB| and no three points of A U B are collinear. Then we show that, the geometric coinpletc bipartite graph K(A, B) contains a spanning tree T without crossings such that the maximum degree of T is at most 3.