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Article: Approximating the chromatic index of multigraphs
Title | Approximating the chromatic index of multigraphs | ||||||||||
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Authors | |||||||||||
Keywords | Chromatic index Edge coloring Fractional chromatic index Multigraph | ||||||||||
Issue Date | 2011 | ||||||||||
Publisher | Springer Verlag Dordrecht. The Journal's web site is located at http://springerlink.metapress.com/openurl.asp?genre=journal&issn=1382-6905 | ||||||||||
Citation | Journal Of Combinatorial Optimization, 2011, v. 21 n. 2, p. 219-246 How to Cite? | ||||||||||
Abstract | It is well known that if G is a multigraph then χ′(G) ≥χ′ *(G):=max{Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ′ *(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max{2|E(G[U])|/(|U|-1):U ⊇ V(G),|U|≥3,|U| is odd}. The conjecture that χ′(G)≤max{Δ(G)+1,Γ(G) was made independently by Goldberg (Discret. Anal. 23:3-7, 1973), Anderson (Math. Scand. 40:161-175, 1977), and Seymour (Proc. Lond. Math. Soc. 38:423-460, 1979). Using a probabilistic argument Kahn showed that for any c>0 there exists D>0 such that χ′(G)≤χ′ *(G)+c χ′ *(G) when χ′ *(G)>D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ′(G)>⌊(11Δ(G)+8)/10⌋ and Scheide recently improved this bound to χ′(G)>⌊(15Δ(G)+12)/14⌋. We prove this conjecture for multigraphs G with χ′(G)> ⌊ Δ(G)+ Δ(G)/2 ⌋, improving the above mentioned results. As a consequence, for multigraphs G with χ(G)>Delta(G)+sqrt Δ(G)/2} the answer to a 1964 problem of Vizing is in the affirmative. © 2009 Springer Science+Business Media, LLC. | ||||||||||
Persistent Identifier | http://hdl.handle.net/10722/137323 | ||||||||||
ISSN | 2021 Impact Factor: 1.262 2020 SCImago Journal Rankings: 0.538 | ||||||||||
ISI Accession Number ID |
Funding Information: G. Chen is partially supported by NSF. | ||||||||||
References |
DC Field | Value | Language |
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dc.contributor.author | Chen, G | en_HK |
dc.contributor.author | Yu, X | en_HK |
dc.contributor.author | Zang, W | en_HK |
dc.date.accessioned | 2011-08-26T14:23:16Z | - |
dc.date.available | 2011-08-26T14:23:16Z | - |
dc.date.issued | 2011 | en_HK |
dc.identifier.citation | Journal Of Combinatorial Optimization, 2011, v. 21 n. 2, p. 219-246 | en_HK |
dc.identifier.issn | 1382-6905 | en_HK |
dc.identifier.uri | http://hdl.handle.net/10722/137323 | - |
dc.description.abstract | It is well known that if G is a multigraph then χ′(G) ≥χ′ *(G):=max{Δ(G),Γ(G)}, where χ′(G) is the chromatic index of G, χ′ *(G) is the fractional chromatic index of G, Δ(G) is the maximum degree of G, and Γ(G)=max{2|E(G[U])|/(|U|-1):U ⊇ V(G),|U|≥3,|U| is odd}. The conjecture that χ′(G)≤max{Δ(G)+1,Γ(G) was made independently by Goldberg (Discret. Anal. 23:3-7, 1973), Anderson (Math. Scand. 40:161-175, 1977), and Seymour (Proc. Lond. Math. Soc. 38:423-460, 1979). Using a probabilistic argument Kahn showed that for any c>0 there exists D>0 such that χ′(G)≤χ′ *(G)+c χ′ *(G) when χ′ *(G)>D. Nishizeki and Kashiwagi proved this conjecture for multigraphs G with χ′(G)>⌊(11Δ(G)+8)/10⌋ and Scheide recently improved this bound to χ′(G)>⌊(15Δ(G)+12)/14⌋. We prove this conjecture for multigraphs G with χ′(G)> ⌊ Δ(G)+ Δ(G)/2 ⌋, improving the above mentioned results. As a consequence, for multigraphs G with χ(G)>Delta(G)+sqrt Δ(G)/2} the answer to a 1964 problem of Vizing is in the affirmative. © 2009 Springer Science+Business Media, LLC. | en_HK |
dc.language | eng | en_US |
dc.publisher | Springer Verlag Dordrecht. The Journal's web site is located at http://springerlink.metapress.com/openurl.asp?genre=journal&issn=1382-6905 | en_HK |
dc.relation.ispartof | Journal of Combinatorial Optimization | en_HK |
dc.rights | The original publication is available at www.springerlink.com | en_US |
dc.subject | Chromatic index | en_HK |
dc.subject | Edge coloring | en_HK |
dc.subject | Fractional chromatic index | en_HK |
dc.subject | Multigraph | en_HK |
dc.title | Approximating the chromatic index of multigraphs | en_HK |
dc.type | Article | en_HK |
dc.identifier.openurl | http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1382-6905&volume=21&issue=2&spage=219&epage=246&date=2011&atitle=Approximating+the+chromatic+index+of+multigraphs | - |
dc.identifier.email | Zang, W:wzang@maths.hku.hk | en_HK |
dc.identifier.authority | Zang, W=rp00839 | en_HK |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1007/s10878-009-9232-y | en_HK |
dc.identifier.scopus | eid_2-s2.0-79851515504 | en_HK |
dc.identifier.hkuros | 190479 | en_US |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-79851515504&selection=ref&src=s&origin=recordpage | en_HK |
dc.identifier.volume | 21 | en_HK |
dc.identifier.issue | 2 | en_HK |
dc.identifier.spage | 219 | en_HK |
dc.identifier.epage | 246 | en_HK |
dc.identifier.isi | WOS:000286680700005 | - |
dc.publisher.place | Netherlands | en_HK |
dc.identifier.scopusauthorid | Chen, G=35794365300 | en_HK |
dc.identifier.scopusauthorid | Yu, X=7404115058 | en_HK |
dc.identifier.scopusauthorid | Zang, W=7005740804 | en_HK |
dc.identifier.citeulike | 4740537 | - |
dc.identifier.issnl | 1382-6905 | - |