On the chromatic number of random d-regular graphs
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d < 2 ( k − 1 ) log ( k − 1 ) . From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to b...
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Published in | Advances in mathematics (New York. 1965) Vol. 223; no. 1; pp. 300 - 328 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
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Elsevier Inc
15.01.2010
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Abstract | In this work we show that, for any fixed
d, random
d-regular graphs asymptotically almost surely can be coloured with
k colours, where
k is the smallest integer satisfying
d
<
2
(
k
−
1
)
log
(
k
−
1
)
. From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely
k
−
1
or
k. If moreover
d
>
(
2
k
−
3
)
log
(
k
−
1
)
, then the value
k
−
1
is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value
k
+
1
. Our proof applies the small subgraph conditioning method to the number of equitable
k-colourings, where a colouring is
equitable if the number of vertices of each colour is equal. |
---|---|
AbstractList | In this work we show that, for any fixed
d, random
d-regular graphs asymptotically almost surely can be coloured with
k colours, where
k is the smallest integer satisfying
d
<
2
(
k
−
1
)
log
(
k
−
1
)
. From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely
k
−
1
or
k. If moreover
d
>
(
2
k
−
3
)
log
(
k
−
1
)
, then the value
k
−
1
is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value
k
+
1
. Our proof applies the small subgraph conditioning method to the number of equitable
k-colourings, where a colouring is
equitable if the number of vertices of each colour is equal. |
Author | Kemkes, Graeme Wormald, Nicholas Pérez-Giménez, Xavier |
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Cites_doi | 10.1007/BF01375472 10.1002/jgt.20369 10.1002/rsa.1013 10.1007/BF01205080 10.1017/S0963548306007954 10.1016/j.disc.2008.12.014 10.1017/S0963548302005254 10.1016/0095-8956(92)90070-E 10.1017/S0963548306007693 10.1007/BF02122551 10.1016/j.jctb.2007.11.009 10.1007/BF01215914 10.4007/annals.2005.162.1335 10.1016/S0195-6698(80)80030-8 10.1002/rsa.3240050209 10.1007/BF02579208 |
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References | Łuczak (bib015) 1991; 11 Achlioptas, Naor (bib003) 2005; 162 Krivelevich, Sudakov, Vu, Wormald (bib014) 2001; 18 Łuczak (bib016) 1991; 11 Shamir, Spencer (bib019) 1987; 7 Shi, Wormald (bib020) 2007; 16 (bib001) 1972; vol. 55 M. Molloy, B.A. Reed, The chromatic number of sparse random graphs, Master's thesis, University of Waterloo, 1992 Cooper, Frieze, Reed, Riordan (bib009) 2002; 11 Robinson, Wormald (bib018) 1994; 5 Wormald (bib022) 1999; vol. 267 de Bruijn (bib010) 1970 Alon, Krivelevich (bib004) 1997; 17 Frieze, Łuczak (bib012) 1992; 54 Bollobás (bib006) 1980; 1 Coja-Oghlan, Panagiotou, Steger (bib008) 2008; 98 Janson, Łuczak, Ruciński (bib013) 2000 Ben-Shimon, Krivelevich (bib005) 2009; 309 Bollobás (bib007) 1988; 8 Achlioptas, Moore (bib002) 2004; vol. 3122 Shi, Wormald (bib021) 2007; 16 Díaz, Kaporis, Kemkes, Kirousis, Pérez, Wormald (bib011) 2009; 61 Wormald (10.1016/j.aim.2009.08.006_bib022) 1999; vol. 267 Alon (10.1016/j.aim.2009.08.006_bib004) 1997; 17 Krivelevich (10.1016/j.aim.2009.08.006_bib014) 2001; 18 Achlioptas (10.1016/j.aim.2009.08.006_bib003) 2005; 162 Robinson (10.1016/j.aim.2009.08.006_bib018) 1994; 5 10.1016/j.aim.2009.08.006_bib017 Bollobás (10.1016/j.aim.2009.08.006_bib007) 1988; 8 Díaz (10.1016/j.aim.2009.08.006_bib011) 2009; 61 Ben-Shimon (10.1016/j.aim.2009.08.006_bib005) 2009; 309 Łuczak (10.1016/j.aim.2009.08.006_bib016) 1991; 11 de Bruijn (10.1016/j.aim.2009.08.006_bib010) 1970 Cooper (10.1016/j.aim.2009.08.006_bib009) 2002; 11 Łuczak (10.1016/j.aim.2009.08.006_bib015) 1991; 11 Shamir (10.1016/j.aim.2009.08.006_bib019) 1987; 7 (10.1016/j.aim.2009.08.006_bib001) 1972; vol. 55 Shi (10.1016/j.aim.2009.08.006_bib021) 2007; 16 Bollobás (10.1016/j.aim.2009.08.006_bib006) 1980; 1 Coja-Oghlan (10.1016/j.aim.2009.08.006_bib008) 2008; 98 Janson (10.1016/j.aim.2009.08.006_bib013) 2000 Shi (10.1016/j.aim.2009.08.006_bib020) 2007; 16 Achlioptas (10.1016/j.aim.2009.08.006_bib002) 2004; vol. 3122 Frieze (10.1016/j.aim.2009.08.006_bib012) 1992; 54 |
References_xml | – volume: 162 start-page: 1333 year: 2005 end-page: 1349 ident: bib003 article-title: The two possible values of the chromatic number of a random graph publication-title: Ann. of Math. contributor: fullname: Naor – volume: 11 start-page: 323 year: 2002 end-page: 341 ident: bib009 article-title: Random regular graphs of non-constant degree: Independence and chromatic number publication-title: Combin. Probab. Comput. contributor: fullname: Riordan – volume: 54 start-page: 123 year: 1992 end-page: 132 ident: bib012 article-title: On the independence and chromatic numbers of random regular graphs publication-title: J. Combin. Theory Ser. B contributor: fullname: Łuczak – volume: 98 start-page: 980 year: 2008 end-page: 993 ident: bib008 article-title: On the chromatic number of random graphs publication-title: J. Combin. Theory Ser. B contributor: fullname: Steger – volume: 16 start-page: 459 year: 2007 end-page: 494 ident: bib021 article-title: Colouring random regular graphs publication-title: Combin. Probab. Comput. contributor: fullname: Wormald – volume: 8 start-page: 49 year: 1988 end-page: 55 ident: bib007 article-title: The chromatic number of random graphs publication-title: Combinatorica contributor: fullname: Bollobás – volume: 5 start-page: 363 year: 1994 end-page: 374 ident: bib018 article-title: Almost all regular graphs are hamiltonian publication-title: Random Structures Algorithms contributor: fullname: Wormald – volume: 11 start-page: 45 year: 1991 end-page: 54 ident: bib015 article-title: The chromatic number of random graphs publication-title: Combinatorica contributor: fullname: Łuczak – volume: 7 start-page: 121 year: 1987 end-page: 129 ident: bib019 article-title: Sharp concentration of the chromatic number on random graphs publication-title: Combinatorica contributor: fullname: Spencer – volume: vol. 55 year: 1972 ident: bib001 publication-title: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables – volume: 11 start-page: 295 year: 1991 end-page: 297 ident: bib016 article-title: A note on the sharp concentration of the chromatic number of random graphs publication-title: Combinatorica contributor: fullname: Łuczak – volume: 1 start-page: 311 year: 1980 end-page: 316 ident: bib006 article-title: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs publication-title: European J. Combin. contributor: fullname: Bollobás – volume: vol. 267 start-page: 239 year: 1999 end-page: 298 ident: bib022 article-title: Models of random regular graphs publication-title: Surveys in Combinatorics 1999 contributor: fullname: Wormald – volume: vol. 3122 start-page: 219 year: 2004 end-page: 228 ident: bib002 article-title: The chromatic number of random regular graphs publication-title: Proc. 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 8th International Workshop on Randomization and Computation contributor: fullname: Moore – year: 2000 ident: bib013 article-title: Random Graphs contributor: fullname: Ruciński – volume: 61 start-page: 157 year: 2009 end-page: 191 ident: bib011 article-title: On the chromatic number of a random 5-regular graph publication-title: J. Graph Theory contributor: fullname: Wormald – volume: 17 start-page: 303 year: 1997 end-page: 313 ident: bib004 article-title: The concentration of the chromatic number of random graphs publication-title: Combinatorica contributor: fullname: Krivelevich – volume: 309 start-page: 4149 year: 2009 end-page: 4161 ident: bib005 article-title: Random regular graphs of non-constant degree: Concentration of the chromatic number publication-title: Discrete Math. contributor: fullname: Krivelevich – year: 1970 ident: bib010 article-title: Asymptotic Methods in Analysis contributor: fullname: de Bruijn – volume: 18 start-page: 346 year: 2001 end-page: 363 ident: bib014 article-title: Random regular graphs of high degree publication-title: Random Structures Algorithms contributor: fullname: Wormald – volume: 16 start-page: 309 year: 2007 end-page: 344 ident: bib020 article-title: Colouring random 4-regular graphs publication-title: Combin. Probab. Comput. contributor: fullname: Wormald – volume: 11 start-page: 45 issue: 1 year: 1991 ident: 10.1016/j.aim.2009.08.006_bib015 article-title: The chromatic number of random graphs publication-title: Combinatorica doi: 10.1007/BF01375472 contributor: fullname: Łuczak – volume: vol. 55 year: 1972 ident: 10.1016/j.aim.2009.08.006_bib001 – year: 1970 ident: 10.1016/j.aim.2009.08.006_bib010 contributor: fullname: de Bruijn – volume: 61 start-page: 157 issue: 3 year: 2009 ident: 10.1016/j.aim.2009.08.006_bib011 article-title: On the chromatic number of a random 5-regular graph publication-title: J. Graph Theory doi: 10.1002/jgt.20369 contributor: fullname: Díaz – volume: 18 start-page: 346 issue: 4 year: 2001 ident: 10.1016/j.aim.2009.08.006_bib014 article-title: Random regular graphs of high degree publication-title: Random Structures Algorithms doi: 10.1002/rsa.1013 contributor: fullname: Krivelevich – volume: 11 start-page: 295 issue: 3 year: 1991 ident: 10.1016/j.aim.2009.08.006_bib016 article-title: A note on the sharp concentration of the chromatic number of random graphs publication-title: Combinatorica doi: 10.1007/BF01205080 contributor: fullname: Łuczak – volume: vol. 3122 start-page: 219 year: 2004 ident: 10.1016/j.aim.2009.08.006_bib002 article-title: The chromatic number of random regular graphs contributor: fullname: Achlioptas – volume: 16 start-page: 459 issue: 3 year: 2007 ident: 10.1016/j.aim.2009.08.006_bib021 article-title: Colouring random regular graphs publication-title: Combin. Probab. Comput. doi: 10.1017/S0963548306007954 contributor: fullname: Shi – volume: vol. 267 start-page: 239 year: 1999 ident: 10.1016/j.aim.2009.08.006_bib022 article-title: Models of random regular graphs contributor: fullname: Wormald – volume: 309 start-page: 4149 issue: 12 year: 2009 ident: 10.1016/j.aim.2009.08.006_bib005 article-title: Random regular graphs of non-constant degree: Concentration of the chromatic number publication-title: Discrete Math. doi: 10.1016/j.disc.2008.12.014 contributor: fullname: Ben-Shimon – volume: 11 start-page: 323 issue: 4 year: 2002 ident: 10.1016/j.aim.2009.08.006_bib009 article-title: Random regular graphs of non-constant degree: Independence and chromatic number publication-title: Combin. Probab. Comput. doi: 10.1017/S0963548302005254 contributor: fullname: Cooper – year: 2000 ident: 10.1016/j.aim.2009.08.006_bib013 contributor: fullname: Janson – volume: 54 start-page: 123 issue: 1 year: 1992 ident: 10.1016/j.aim.2009.08.006_bib012 article-title: On the independence and chromatic numbers of random regular graphs publication-title: J. Combin. Theory Ser. B doi: 10.1016/0095-8956(92)90070-E contributor: fullname: Frieze – volume: 16 start-page: 309 issue: 2 year: 2007 ident: 10.1016/j.aim.2009.08.006_bib020 article-title: Colouring random 4-regular graphs publication-title: Combin. Probab. Comput. doi: 10.1017/S0963548306007693 contributor: fullname: Shi – volume: 8 start-page: 49 issue: 1 year: 1988 ident: 10.1016/j.aim.2009.08.006_bib007 article-title: The chromatic number of random graphs publication-title: Combinatorica doi: 10.1007/BF02122551 contributor: fullname: Bollobás – volume: 98 start-page: 980 issue: 5 year: 2008 ident: 10.1016/j.aim.2009.08.006_bib008 article-title: On the chromatic number of random graphs publication-title: J. Combin. Theory Ser. B doi: 10.1016/j.jctb.2007.11.009 contributor: fullname: Coja-Oghlan – volume: 17 start-page: 303 issue: 3 year: 1997 ident: 10.1016/j.aim.2009.08.006_bib004 article-title: The concentration of the chromatic number of random graphs publication-title: Combinatorica doi: 10.1007/BF01215914 contributor: fullname: Alon – ident: 10.1016/j.aim.2009.08.006_bib017 – volume: 162 start-page: 1333 issue: 3 year: 2005 ident: 10.1016/j.aim.2009.08.006_bib003 article-title: The two possible values of the chromatic number of a random graph publication-title: Ann. of Math. doi: 10.4007/annals.2005.162.1335 contributor: fullname: Achlioptas – volume: 1 start-page: 311 issue: 4 year: 1980 ident: 10.1016/j.aim.2009.08.006_bib006 article-title: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs publication-title: European J. Combin. doi: 10.1016/S0195-6698(80)80030-8 contributor: fullname: Bollobás – volume: 5 start-page: 363 issue: 2 year: 1994 ident: 10.1016/j.aim.2009.08.006_bib018 article-title: Almost all regular graphs are hamiltonian publication-title: Random Structures Algorithms doi: 10.1002/rsa.3240050209 contributor: fullname: Robinson – volume: 7 start-page: 121 issue: 1 year: 1987 ident: 10.1016/j.aim.2009.08.006_bib019 article-title: Sharp concentration of the chromatic number on random graphs Gn,p publication-title: Combinatorica doi: 10.1007/BF02579208 contributor: fullname: Shamir |
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Snippet | In this work we show that, for any fixed
d, random
d-regular graphs asymptotically almost surely can be coloured with
k colours, where
k is the smallest... |
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Title | On the chromatic number of random d-regular graphs |
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