We prove the following graph coloring result: Let G be a 2{connected bipartite planar graph. Then one can triangulate G in such a w ay that the resulting graph is 3{colorable. This result implies several new upper bounds for guarding problems including the rst non{trivial upper bound for the rectilinear Prison Yard Problem: 1. n 3 vertex guards are su cient t o w atch the interior of a rectilinear polygon with holes. 2. 5n 12 + 3 v ertex guards resp. n+4 3 point guards are su cient t o w atch simultaneously both the interior and exterior of a rectilinear polygon. Moreover, we s h o w a new lower bound of 5n 16 vertex guards for the rectilinear Prison Yard Problem and prove it to be asymptotically tight for the class of orthoconvex polygons.
Hoffmann and Kriegel showed that a 2-connected, bipartite, planar graph can be triangulated such that the resulting graph is 3-colorable REF , implying even degrees.
A Graph-Coloring Result and Its Consequences For Polygon-Guarding Problems
SIAM J. Discrete Math.
SIAM J. Discrete Math.
Mathematics Computer Science