Odd components of co-trees and graph embeddings

Han Ren and Fugang Chao

Applied Mathematics and Computation, 2016, vol. 286, issue C, 228-231

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=12(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 [9,13] to some other (possible) genera. Infinitely many examples show that there are graphs which satisfy (a) but (b). Those make a correction of a result of Archdeacon [2, Theorem 1].

Keywords: Odd component of co-trees; Graph embeddings; Spanning tree (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:286:y:2016:i:c:p:228-231

DOI: 10.1016/j.amc.2016.04.027

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