Flows, flow-pair covers and cycle double covers
type: Item
instance of: scholarly article
title: Flows, flow-pair covers and cycle double covers
zbMATH Open document ID: 1218.05156
author: Cun-Quan Zhang
author: Dezheng Xie
published in: Discrete Mathematics
publication date: 2009-12-15T00:00:00Z
review text: In this paper, some earlier results by \textit{H. Fleischner} [``Bipartizing matchings and Sabidussi's compatibility conjecture'', Discrete Math. 244, No. 1--3, 77--82 (2002; Zbl 0989.05063)] about edge-disjoint bipartizing matchings of a cubic graph with a dominating circuit are generalized for graphs without the assumption of the existence of a dominating circuit and 3-regularity. A pair of integer flows \((D,f_{1})\) and \((D,f_{2})\) is an \((h,k)\)-flow parity-pair-cover of \(G\) if the union of their supports covers the entire graph; \(f_{1}\) is an \(h\)-flow and \(f_{2}\) is a \(k\)-flow, and \(E_{f_1=\mathrm{odd}}=E_{f_2=\mathrm{odd}}\). Then \(G\) admits a nowhere-zero 6-flow if and only if \(G\) admits a \((4,3)\)-flow parity-pair-cover; and \(G\) admits a nowhere-zero 5-flow if \(G\) admits a \((3,3)\)-flow parity-pair-cover. A pair of integer flows \((D,f_{1})\) and \((D,f_{2})\) is an \((h,k)\)-flow even-disjoint-pair-cover of \(G\) if the union of their supports covers the entire graph, \(f_{1}\) is an \(h\)-flow and \(f_{2}\) is a \(k\)-flow, and \(E_{f_i=\mathrm{even},f_i\neq 0}\subseteq E_{f_j=0}\) for each \(\{i,j\}=\{1,2\}\). Then \(G\) has a 5-cycle double cover if \(G\) admits a \((4,4)\)-flow even-disjoint-pair-cover; and \(G\) admits a \((3,3)\)-flow parity-pair-cover if \(G\) has an orientable 5-cycle double cover.
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 5648010
zbMATH Keywords: integer flow
zbMATH Keywords: flow cover
zbMATH Keywords: cycle double cover
MaRDI profile type: Publication
full work available at URL: j.disc.2008.05.056
OpenAlex ID: W1971381012
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DOI: 10.1016/J.DISC.2008.05.056
description: scientific article; zbMATH DE number 5648010