CW>KimMarshall: New page: ===''ex(n;4)''=== P.Erdös P. Erdös, Some recent progress on the extremal problems in graph theory, ''Congr. Numer.'' '''14''' (1975) 3--14 conjectured that ''ex(n;4...

2011-02-28T06:45:38Z

New page: ===''ex(n;4)''=== P.Erdös <ref>P. Erdös, Some recent progress on the extremal problems in graph theory, ''Congr. Numer.'' '''14''' (1975) 3--14</ref> conjectured that ''ex(n;4...

New page

===''ex(n;4)''===

P.Erdös <ref>P. Erdös, Some recent progress on the extremal problems in graph theory, ''Congr. Numer.'' '''14''' (1975) 3--14</ref> conjectured that ''ex(n;4)=(1/2 + o(1))3/2 n3/2''.

Garnick ''et al''. demonstrated that extremal graphs include:
* Stars ''K1,n-1'' and Paths ''Pn'' for ''n''  ≤  ''4'',
* [http://en.wikipedia.org/wiki/Moore_graph Moore graphs]:
** ''EX(5;4)={C5}'' the cycle;
** ''ex(10;4)= 15'' is attained by the [http://en.wikipedia.org/wiki/Petersen_graph Petersen graph] ;
** ''ex(50;4)=175'' is attained by the [http://en.wikipedia.org/wiki/Hoffman-Singleton_graph Hoffman-Singleton graph];
** ''ex(3250;4)'' is attained by the Moore graph with girth 5 and degree 57 if it exists.
* ''C6'' is an element of ''EX(6;4)''.
* polarity graphs.

Let ''G'' be an extremal graph of order ''n''. Then
# the diameter of ''G'' is at most 3;
# if ''d(x)=δ(G)=1, then the graph ''G-{x}'' has diameter at most 2.

The following table lists the known values and constructive lower bounds for ''ex(n;4)''. If the value is known then it is shown in bold text.

{|style="border-collapse: separate; border-spacing: 0; border-width: 1px; border-style: solid; border-color: #000; padding: 0" cellpadding="10" cellspacing="0"
|+ Known values and constructive lower bounds for ''ex(n;4)''
|-
!style="border-style: solid; border-width: 0 1px 1px 0; background-color: #cccccc;"| ''n''
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 0
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 1
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 2
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 3
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 4
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 5
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 6
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 7
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 8
!style="border-style: solid; border-width: 0 0 1px 0; background-color: #cccccc;"| 9

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 0
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''0'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''0'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''1'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''2'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''3'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''5'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''6'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''8'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''10'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''12'''

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 10
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''15'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''16'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''18'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''21'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''23'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''26'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''28'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''31'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''34'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''38'''

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| '''20'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''41'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''44'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''47'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''50'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| '''54'''
|style="border-style: solid; border-width: 0" style="background-color: #c69aab;"| '''57'''
|style="border-style: solid; border-width: 0" style="background-color: #c69aab;"| '''61'''
|style="border-style: solid; border-width: 0" style="background-color: #c69aab;"| '''65'''
|style="border-style: solid; border-width: 0"| '''68'''
|style="border-style: solid; border-width: 0"| '''72'''

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 30
|style="border-style: solid; border-width: 0"| '''76'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 80
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 85
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 87
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 90
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 94
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 99
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 104
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 109
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 114

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 40
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 120
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 124
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 129
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 134
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 139
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 144
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 150
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 156
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 162
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 168

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 50
|style="border-style: solid; border-width: 0"| '''175'''
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 176
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 178
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 181
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 185
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 188
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 192
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 195
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 199
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 203

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 60
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 207
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 212
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 216
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 221
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 226
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 231
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 235
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 240
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 245
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 250

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 70
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 255
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 260
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 265
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 270
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 280
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 291
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 296
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 301

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!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 80
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 306
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 311
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 317
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 323
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 329
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 334
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 340
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 346
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 352
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 357

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 90
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 363
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 368
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 374
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 379
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 385
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 391
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 398
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 404
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 410
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 416

|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 100
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 422
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 428
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 434
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 440
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 446
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 452
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 458
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 464
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 470
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 476

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!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 110
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 483
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 489
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 495
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 520
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 538

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!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 120
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 544
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 551
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 558
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 565
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 571
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 578
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 584
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!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 130
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 610
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 617
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 623
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 630
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 637
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 644
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 651
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 658
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 665
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 672

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!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 140
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 679
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 686
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 693
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 700
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 749
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 763
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 770
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 777
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 784
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 791
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 798
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 819
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|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 1113
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 1121
|-
!style="border-style: solid; border-width: 0 1px 0 0; background-color: #cccccc;"| 200
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"| 1129
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|style="border-style: solid; border-width: 0" style="background-color: #d9ddc3;"|
|}

The following table is the key to the colors in the table presented above:

{| border="1" cellspacing="1" cellpadding="1" style="text-align: left;"
|'''Color''' || style="text-align: center;" |'''References'''
|-
|style="background-color: #d9ddc3; text-align: center;" | * || Garnick, Kwong and Lazebnik<ref>D. K. Garnick and Y. H. H. Kwong, F. Lazebnik, Extremal graphs without three-cycles or four-cycles, '' J. Graph Theory'', ''' 17''' (1993), 633--645</ref>.
|-
|style="background-color: #c69aab; text-align: center;" | * || Garnick and Nieuwejaar<ref>D. K. Garnick and N.A. Nieuwejaar, Non-isomorphic extremal graphs without three-cycles or four-cycles, '' JCMCC'', ''' 12''' (1992), 33--56</ref>.
|-
|}

Girth greater than 4 - Revision history

Grahame: 1 revision imported

CW>KimMarshall: New page: ===''ex(n;4)''=== P.Erdös P. Erdös, Some recent progress on the extremal problems in graph theory, ''Congr. Numer.'' '''14''' (1975) 3--14 conjectured that ''ex(n;4...

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