Finding the exact bound of the maximum degrees of class two graphs embeddable in a surface of characteristic ε∈{-1,-2,-3}

Authors:
Rong Luo;Yue Zhao
Affiliations:
Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37130, USA;Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2008

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Abstract

In this paper, we consider the problem of determining the maximum of the set of maximum degrees of class two graphs that can be embedded in a surface. For each surface @S, we define @D(@S)=max{@D(G)|G is a class two graph of maximum degree @D that can be embedded in @S}. Hence Vizing's Planar Graph Conjecture can be restated as @D(@S)=5 if @S is a plane. We show that @D(@S)=7 if @e(@S)=-1 and @D(@S)=8 if @e(@S)@?{-2,-3}.