On the Grundy and b-Chromatic Numbers of a Graph
The Grundy number of a graph G , denoted by Γ ( G ), is the largest k such that G has a greedy k -colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertexes of G . The b - chromatic number of a graph G , denoted by χ b ( G ), is...
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| Veröffentlicht in: | Algorithmica 2013-04, Vol.65 (4), p.885-899 |
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| container_title | Algorithmica |
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| creator | Havet, Frédéric Sampaio, Leonardo |
| description | The
Grundy number
of a graph
G
, denoted by
Γ
(
G
), is the largest
k
such that
G
has a
greedy
k
-colouring, that is a colouring with
k
colours obtained by applying the greedy algorithm according to some ordering of the vertexes of
G
. The
b
-
chromatic number
of a graph
G
, denoted by
χ
b
(
G
), is the largest
k
such that
G
has a
b
-
colouring
with
k
colours, that is a colouring in which each colour class contains a
b
-
vertex
, a vertex with neighbours in all other colour classes. Trivially
χ
b
(
G
),
Γ
(
G
)≤
Δ
(
G
)+1. In this paper, we show that deciding if
Γ
(
G
)≤
Δ
(
G
) is NP-complete even for a bipartite graph
G
. We then show that deciding if
Γ
(
G
)≥|
V
(
G
)|−
k
or if
χ
b
(
G
)≥|
V
(
G
)|−
k
are fixed parameter tractable problems with respect to the parameter
k
. |
| doi_str_mv | 10.1007/s00453-011-9604-4 |
| format | Article |
| fullrecord | <record><control><sourceid>hal_cross</sourceid><recordid>TN_cdi_hal_primary_oai_HAL_hal_00905927v1</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>oai_HAL_hal_00905927v1</sourcerecordid><originalsourceid>FETCH-LOGICAL-c431t-630a3836fefaa24dcc393b115b18615f91b610652d92eaa2ba4447b22de845ef3</originalsourceid><addsrcrecordid>eNp9kE1LAzEQQIMoWKs_wFuuHqIz-drNsRS1QrEXPYdkN3Fb2t2StEL_vSkrHj0NDO8NzCPkHuERAaqnDCCVYIDIjAbJ5AWZoBScgZJ4SSaAVc2kxuqa3OS8AUBeGT0hsOrpoQv0NR379kRd31LP5l0adu6wbuj7cedDynSI1BXG7btbchXdNoe73zklny_PH_MFW65e3-azJWukwAPTApyohY4hOsdl2zTCCI-oPNYaVTToNYJWvDU8FMI7KWXlOW9DLVWIYkoexrud29p9Wu9cOtnBre1itrTnHYABZXj1jYXFkW3SkHMK8U9AsOc8dsxjSx57zmNlcfjo5ML2XyHZzXBMfXnpH-kHrXNkyA</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>On the Grundy and b-Chromatic Numbers of a Graph</title><source>SpringerNature Journals</source><creator>Havet, Frédéric ; Sampaio, Leonardo</creator><creatorcontrib>Havet, Frédéric ; Sampaio, Leonardo</creatorcontrib><description>The
Grundy number
of a graph
G
, denoted by
Γ
(
G
), is the largest
k
such that
G
has a
greedy
k
-colouring, that is a colouring with
k
colours obtained by applying the greedy algorithm according to some ordering of the vertexes of
G
. The
b
-
chromatic number
of a graph
G
, denoted by
χ
b
(
G
), is the largest
k
such that
G
has a
b
-
colouring
with
k
colours, that is a colouring in which each colour class contains a
b
-
vertex
, a vertex with neighbours in all other colour classes. Trivially
χ
b
(
G
),
Γ
(
G
)≤
Δ
(
G
)+1. In this paper, we show that deciding if
Γ
(
G
)≤
Δ
(
G
) is NP-complete even for a bipartite graph
G
. We then show that deciding if
Γ
(
G
)≥|
V
(
G
)|−
k
or if
χ
b
(
G
)≥|
V
(
G
)|−
k
are fixed parameter tractable problems with respect to the parameter
k
.</description><identifier>ISSN: 0178-4617</identifier><identifier>EISSN: 1432-0541</identifier><identifier>DOI: 10.1007/s00453-011-9604-4</identifier><language>eng</language><publisher>New York: Springer-Verlag</publisher><subject>Algorithm Analysis and Problem Complexity ; Algorithms ; Computer Science ; Computer Systems Organization and Communication Networks ; Data Structures and Information Theory ; Discrete Mathematics ; Mathematics of Computing ; Theory of Computation</subject><ispartof>Algorithmica, 2013-04, Vol.65 (4), p.885-899</ispartof><rights>Springer Science+Business Media, LLC 2011</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c431t-630a3836fefaa24dcc393b115b18615f91b610652d92eaa2ba4447b22de845ef3</citedby><cites>FETCH-LOGICAL-c431t-630a3836fefaa24dcc393b115b18615f91b610652d92eaa2ba4447b22de845ef3</cites><orcidid>0000-0002-3447-8112</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00453-011-9604-4$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00453-011-9604-4$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,27924,27925,41488,42557,51319</link.rule.ids><backlink>$$Uhttps://inria.hal.science/hal-00905927$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Havet, Frédéric</creatorcontrib><creatorcontrib>Sampaio, Leonardo</creatorcontrib><title>On the Grundy and b-Chromatic Numbers of a Graph</title><title>Algorithmica</title><addtitle>Algorithmica</addtitle><description>The
Grundy number
of a graph
G
, denoted by
Γ
(
G
), is the largest
k
such that
G
has a
greedy
k
-colouring, that is a colouring with
k
colours obtained by applying the greedy algorithm according to some ordering of the vertexes of
G
. The
b
-
chromatic number
of a graph
G
, denoted by
χ
b
(
G
), is the largest
k
such that
G
has a
b
-
colouring
with
k
colours, that is a colouring in which each colour class contains a
b
-
vertex
, a vertex with neighbours in all other colour classes. Trivially
χ
b
(
G
),
Γ
(
G
)≤
Δ
(
G
)+1. In this paper, we show that deciding if
Γ
(
G
)≤
Δ
(
G
) is NP-complete even for a bipartite graph
G
. We then show that deciding if
Γ
(
G
)≥|
V
(
G
)|−
k
or if
χ
b
(
G
)≥|
V
(
G
)|−
k
are fixed parameter tractable problems with respect to the parameter
k
.</description><subject>Algorithm Analysis and Problem Complexity</subject><subject>Algorithms</subject><subject>Computer Science</subject><subject>Computer Systems Organization and Communication Networks</subject><subject>Data Structures and Information Theory</subject><subject>Discrete Mathematics</subject><subject>Mathematics of Computing</subject><subject>Theory of Computation</subject><issn>0178-4617</issn><issn>1432-0541</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQQIMoWKs_wFuuHqIz-drNsRS1QrEXPYdkN3Fb2t2StEL_vSkrHj0NDO8NzCPkHuERAaqnDCCVYIDIjAbJ5AWZoBScgZJ4SSaAVc2kxuqa3OS8AUBeGT0hsOrpoQv0NR379kRd31LP5l0adu6wbuj7cedDynSI1BXG7btbchXdNoe73zklny_PH_MFW65e3-azJWukwAPTApyohY4hOsdl2zTCCI-oPNYaVTToNYJWvDU8FMI7KWXlOW9DLVWIYkoexrud29p9Wu9cOtnBre1itrTnHYABZXj1jYXFkW3SkHMK8U9AsOc8dsxjSx57zmNlcfjo5ML2XyHZzXBMfXnpH-kHrXNkyA</recordid><startdate>20130401</startdate><enddate>20130401</enddate><creator>Havet, Frédéric</creator><creator>Sampaio, Leonardo</creator><general>Springer-Verlag</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0002-3447-8112</orcidid></search><sort><creationdate>20130401</creationdate><title>On the Grundy and b-Chromatic Numbers of a Graph</title><author>Havet, Frédéric ; Sampaio, Leonardo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c431t-630a3836fefaa24dcc393b115b18615f91b610652d92eaa2ba4447b22de845ef3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>Algorithm Analysis and Problem Complexity</topic><topic>Algorithms</topic><topic>Computer Science</topic><topic>Computer Systems Organization and Communication Networks</topic><topic>Data Structures and Information Theory</topic><topic>Discrete Mathematics</topic><topic>Mathematics of Computing</topic><topic>Theory of Computation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Havet, Frédéric</creatorcontrib><creatorcontrib>Sampaio, Leonardo</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Algorithmica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Havet, Frédéric</au><au>Sampaio, Leonardo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Grundy and b-Chromatic Numbers of a Graph</atitle><jtitle>Algorithmica</jtitle><stitle>Algorithmica</stitle><date>2013-04-01</date><risdate>2013</risdate><volume>65</volume><issue>4</issue><spage>885</spage><epage>899</epage><pages>885-899</pages><issn>0178-4617</issn><eissn>1432-0541</eissn><abstract>The
Grundy number
of a graph
G
, denoted by
Γ
(
G
), is the largest
k
such that
G
has a
greedy
k
-colouring, that is a colouring with
k
colours obtained by applying the greedy algorithm according to some ordering of the vertexes of
G
. The
b
-
chromatic number
of a graph
G
, denoted by
χ
b
(
G
), is the largest
k
such that
G
has a
b
-
colouring
with
k
colours, that is a colouring in which each colour class contains a
b
-
vertex
, a vertex with neighbours in all other colour classes. Trivially
χ
b
(
G
),
Γ
(
G
)≤
Δ
(
G
)+1. In this paper, we show that deciding if
Γ
(
G
)≤
Δ
(
G
) is NP-complete even for a bipartite graph
G
. We then show that deciding if
Γ
(
G
)≥|
V
(
G
)|−
k
or if
χ
b
(
G
)≥|
V
(
G
)|−
k
are fixed parameter tractable problems with respect to the parameter
k
.</abstract><cop>New York</cop><pub>Springer-Verlag</pub><doi>10.1007/s00453-011-9604-4</doi><tpages>15</tpages><orcidid>https://orcid.org/0000-0002-3447-8112</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
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| ispartof | Algorithmica, 2013-04, Vol.65 (4), p.885-899 |
| issn | 0178-4617 1432-0541 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_00905927v1 |
| source | SpringerNature Journals |
| subjects | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Discrete Mathematics Mathematics of Computing Theory of Computation |
| title | On the Grundy and b-Chromatic Numbers of a Graph |
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