...ACES AND SMARANDACHE GEOMETRIES American Research Press Rehoboth, NM 2005 Linfan MAO Institute of Systems Science Academy of Mathematics and Systems ... ...D SMARANDACHE GEOMETRIES American Research Press Rehoboth, NM 2005 Linfan MAO Institute of Systems Science Academy of Mathematics and Systems Chinese... ...ty, Mansfield, PA 16933, USA. Copyright 2005 by American Research Press and Linfan Mao Rehoboth, Box 141 NM 87322, USA Many books can be downloaded fro... ...sfield, PA 16933, USA. Copyright 2005 by American Research Press and Linfan Mao Rehoboth, Box 141 NM 87322, USA Many books can be downloaded from the f... ...l presented here. Any suggestions for improving this book are welcome. L.F.Mao AMSS, Beijing June, 2005 Contents Preface................................ ...ometry of Curves and Surfaces, Pearson Education asia Ltd (2004). [43] L.F.Mao, A census of maps on surface with given underlying graph, A doctor thes... ..., A doctor thesis in Northern Jiaotong University, Beijing, 2002. [44] L.F.Mao and Y.P.Liu, New automorphism groups identity of trees, Chinese Advance... ...ormization theorem 104 voltage map 33 walk 5 y-linear 24 About the Author LinfanMaoisapost-doctorintheAcademyofMathematicsandSystemsofChinese Academy...

A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combinatorial configuration with the classical mathematics, it fascinates more and more mathematician’s interesting. Its function and role in mathematics are widely accepted by mathematicians today. On the last century, many works are concentrated on the combinatorial properties of maps. The main trend is the enumeration of maps, particularly the rooted maps, pioneered by W. Tutte, and today, this kind of papers are still appeared on the journals frequently today. All of those is surveyed in Liu’s book [33]. To determine the embedding of a graph on surfaces, including coloring a map on surfaces is another trend in map theory. Its object is combinatorialization of surfaces, see Gross and Tucker [22], Mohar and Thomassen [53] and White [70], especially the [53] for detail. The construction of regular maps on surfaces, related maps with groups and geometry is a glimmer of the map theory with other mathematics....