Hamiltonian Cycles in Polyhedral Maps

We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of contractible Hamiltonian cycle in equivelar triangulated maps. We also present an algorithm to construct such cycles whenever it exists.Comment: 14 page

Tutte trails of graphs on surfaces

PhDA Tutte trail T of a graph G is a trail such that every component of GnV (T) has at most three edges connecting it to T. In 1992, Bill Jackson conjectured that every 2-edge-connected graph G has a Tutte closed trail. In this thesis, we show that Jackson's conjecture is true when G is embedded on the plane and the projective plane. We also give some partial results when G is embedded on the torus.Department of Mathematics, Ramkhamhaeng University, and Thai governmen

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version