@MISC{Ren_oddcomponents, author = {Han Ren and Dengju Ma and Junjie Lu}, title = {Odd Components of Co-Trees and Graph Embeddings 1}, year = {} }

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Abstract

Abstract: In this paper we investigate the relation between odd components of co-trees and graph embeddings. We show that any graph G must share one of the following two conditions: (a) for each integer h such that G may be embedded on Sh, the sphere with h handles, there is a spanning tree T in G such that h = 1 2 (β(G)−ω(T)), where β(G) and ω(T) are, respectively, the Betti number of G and the number of components of G − E(T) having odd number of edges; (b) for every spanning tree T of G, there is an orientable embedding of G with exact ω(T) + 1 faces. This extends Xuong and Liu’s theorem[5,6] to some other ( possible) genera. Infinitely many examples show that there are graphs which satisfy (a) but (b). Those make a correction of a rseult of D.Archdeacon[2, theorem 1].