On solutions of Alspach's problems
type: Item
instance of: scholarly article
title: On solutions of Alspach's problems
zbMATH Open document ID: 0808.05081
DOI: 10.1007/BF03182611
published in: Chinese Science Bulletin
publication date: 1995-03-07T00:00:00Z
review text: Let \(G\) be a graph and \(a \leq b\) integers. A spanning subgraph of \(G\) is called an \([a, b]\)-factor if for every vertex \(v\) we have \(a \leq d_ h (v) \leq b\), where \(d_ H(v)\) denotes the degree of vertex \(v\) in \(H\). Similarly are defined \([a, b]\)-graphs. The author solves the problem of the decomposition of a graph with given matching \(H\) into \([a, b]\)- factors \(F_ 1,\dots ,F_ m\) orthogonal to \(H\). The factorisation \(F=\{F_ 1,\dots ,F_ m\}\) is orthogonal to the matching \(H\) if \(| E (H) \cap E (F_ i) |= 1\) for \(i=1,\dots ,m\). It is proved that if \(G\) is an \([ma-1, mb+1]\)-graph then for an arbitrary matching \(H\) of \(G\) there exists an \([a-b, b+1]\)-factorisation orthogonal to \(H\). In some sense it is the best possible result as there exists a \([ma, mb]\)-graph with matching \(H\) not orthogonal to any factorisation of \(G\).
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 640466
zbMATH Keywords: Alspach's problems
zbMATH Keywords: \([a, b]\)-graph
zbMATH Keywords: decomposition
zbMATH Keywords: matching
zbMATH Keywords: factorisation
author: Guizhen Liu
reviewed by: Ludovit Niepel
MaRDI profile type: Publication
cites work: Q4212440
cites work: Q3755537
full work available at URL: https://doi.org/10.1007/bf03182611
OpenAlex ID: W2116828261
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description: scientific article; zbMATH DE number 640466