Te???a G??f?? Te

1
Te???a G??f??Teµe???se??-???????µ??-?fa?µ?????e
f??a?? 6 ???µat?sµ??
2
???µat?sµ?? ????f??

• ???µat?sµ?? coloring e??? ???f?? e??a? ?
  ap?d?s? ???µ?t?? st?? ????f?? t??, ?ste ?a µ??
  ?p?????? ?e?t?????? ????f?? µe t? ?d?? ???µa
• ???µat??? t??? -color class- e??a? t? s?????
  ????f?? µe t? ?d?? ???µa
• ?p?s?? d?a??????µe ???µat?sµ? a?µ?? ?a?
  ???µat?sµ? pe??????

3
?a??de??µa

• ? ep?µe??? ???f?? C7 ??e? ???µat?ste? µe 3
  ???µata. ???ste?? ???s?µ?p?????ta? ?? ???µat????
  t??e?? v1,v3,v5,v2,v4,v6,v7, e?? de??? ??
  ???µat???? t??e?? v1,v4,v2,v5,v7,v3,v6

4
???µat???? ????µ??

• ??a ???f?? ????? ß?????? ???eta? k-???µat?s?µ??
  (k-colorable), a? ?? ????f?? t?? µp????? ?a
  ???µat?s???? µe k t? p??? ???µata
• G??f?? k-???µat???? (k-chromatic) ?? ????f?? t??
  µp????? ?a ???µat?s???? µe k ???µata, a??? ??? µe
  k1. ??a, ??a? k-???µat???? de? e??a?
  (k1)-???µat?s?µ??
• ???µat???? a???µ?? (chromatic number) e??a? ?
  µ????te??? a???µ?? ???µ?t?? ?(G)k p??
  ??e?????ta? ??a ?a ???µat?s??µe t?? ????f?? e???
  ???f??, ?ts? ?ste 2 ?e?t?????? ????f?? t?? ?a
  ????? d?af??et??? ???µa

5
???µat???? ????µ??

• ?? ??a? ???f?? G e??a? k-???µat?s?µ??, t?te
  ?(G)k, e?? a? e??a? k-???µat???? t?te ?(G)k
• ?? ??a ???f?? G e??a? p-???µat?s?µ?? (pgt?(G)),
  t?te ? ???f?? G e??a? ?a? r-???µat?s?µ??
  (pgtrgt?(G))

6
???ad??? ???µat?s?µ?? G??f??

• G??f?? ???µat?s?µ?? ?at? µ??ad??? t??p? -uniquely
  colorable- e??a? e?e???? p?? ??e? ??a
  s???e???µ??? s????? ???µat???? t??e??, ????? ?a
  e??a? d??at? ?a ?atas?e?aste? ??a d?af??et???
  s?????.
• ? ep?µe??? ???f?? ??e? ???µat???? t??e?? v1,
  v2,v4, v3,v5

7
???s?µ?? G??f??

• ???s?µ?? (critical) ???f?? ?(?)lt?(G), ??a ???e
  ??G
• k-???s?µ?? (k-critical) ???f?? o G e??a?
  ???s?µ?? ?a? k-???µat????
• Te???µa ?? ? G e??a? k-???s?µ??, t?te d(G)k1

4-???s?µ?? ???f??
8
???e??? G??f??

• ????a clique- e??? ???f?? G e??a? o µ???st??
  p????? ?p????f?? H t?? G.
• T??? order- t?? ????a? ???eta? a???µ?? t??
  ????a? ?(G)
• ???e??? perfect- ???f?? a???µ?? ????a?
  ?(G)?(G)
• ? ep?µe??? t??e??? ???f?? ??e? ?(G)3, e?? ????a
  t?? G e??a? ? ?3 µe a???µ? ????a? ?(G)3

?? d????f?? e??a? t??e???
9
?e??pt?se?? ???µat?sµ??

• ?(Kn)n.
• ?(?n)1.
• ?(Km,n)2.
• ?(?)2.
• ?(G)2, a? ? G?Nn de? pe????e? ????? pe??tt??
  µ?????.

??at?
10
???µat?sµ?? ??????

• ?(C2n)2
• ?(C2n1)3

11
?????e?d?? G??f??

• ? t????e?d?? ???f?? Wn pe????e? ??a ????? t????
  n-1 (Cn-1) ?a? ???e ????f? a?t?? t?? ??????
  s??d?eta? µe µ?a ???? ????f? p?? ???µ??eta?
  ??µß??? s?µe?? (hub).
• ?? a?µ?? t?? ???f?? p?? pe??????? t? ??µß???
  s?µe?? ?????ta? a?t??e? (spokes)

12
???µat???? ????µ?? t?? Wn

• ?(W2n)4
• ?(W2n1)3

13
??µe??? G??f??

• ???e 2-???µat???? ???f?? e??a? d?µe???, ??at? ??
  ????f?? t?? µp????? ?a d?a????s???? se 2
  ?p?s????a V1 ?a? V2, ?ts? ?ste ?a µ?? ?p???e?
  a?µ? p?? ?a e???e? ????f?? t?? ?d??? s??????.
• ???e d?µe??? ???f?? de? e??a? 2-???µat????. ?.?
  ? µ?de????? ???f?? Nn µe n ????f?? ?a? m0 a?µ??
  µp??e? ?a ?e????e? d?µe??? a??? e??a? 1-???µat????

14
??p??a Te???µata

• Te???µa ???e ap??? ???f?? µ???st?? ßa?µ??
  ????f?? D e??a? (D1)-???µat?s?µ??, d??ad?
  ?(G)D1
• Te???µa (Brooks 1941) ???e ap??? ?a?
  s??dedeµ???? ???f?? G?Kn µe D(G)3 e??a?
  D-???µat?s?µ??, d??ad? ?(G)D
• Te???µa (Brooks 1941 - ???? d?at?p?s?) ???e
  ap??? ?a? s??dedeµ???? ???f?? G?Kn p?? de? e??a?
  ???????? ???f?? pe??tt?? µ????? (G?C2n1) e??a?
  D-???µat?s?µ??, d??ad? ?(G)D

15
??? ???a ???µat???? ????µ??

• ???t?µa ????? e??a? ? ???µat???? a???µ?? e???
  ???f?? G
• ?p??t?s? r?(G)n, a? ?p???e? ?p????f?? Kr
• ???a?? ??? ???a t?? ?(G)
• n, o a???µ?? t?? ????f?? t?? G
• D1, ?p?? D ? µ???st?? ßa?µ?? t?? ????f?? t?? G
  (Te???µa)
• D, ?p?? D e??a? ? µ???st?? ßa?µ?? t?? ????f?? t??
  G (Te???µa Brooks) µe t?? p???p??es? ?t? G?Kn ?
  Cn (??a pe??tt? n)

16
??t? ???a ???µat???? ????µ??

• ???a?? ??t? ???a t?? ?(G)
• ? t??? t?? µ???st?? ????a? Kr t?? G
• ?? ?e???µa t?? Brooks de? d??e? p??ta ??a ?a??
  ep??? ????.
• ?st? ? p????? d?µe??? ???f?? K1,12
  (aste??e?d??). ?p? t? ?e???µa Brooks p????pte?
  ?(G)12, e?? ?s??e? ?(G)2

17
?s??s? 1

• ?a p??sd????ste? ? ???µat???? a???µ?? ??a t??
  ep?µe?? ???f?
• ??s? ? G pe????e? t?? K4, ??a ?(G)4. ??p?a
  de???eta? ??a? ???µat?sµ?? t?? G µe 4 ???µata.
  ??a ?(G)4. S??ep?? ?(G)4

18
?s??s? 2

• G?a t?? ep?µe?? ???f? G, ?p?????ste
• ?? µ????te?? t?µ? t?? ?(G) µe ß?s? t?? t??? t??
  µe?a??te??? p?????? ?p????f??
• ?? µe?a??te?? t?µ? t?? ?(G) µe ß?s? t? Te???µa
  Brooks
• ??? te???? t?µ? t?? ?(G)k ?a??? ?a? ??a?
  k-???µat?sµ? t?? G

19
?s??s? 2

• ??s? ?pe?d? pe????eta? ? p????? ???f?? K3 ?
  ?(G)3. ?p? Brooks ? ?(G) D3. ??µ? p??
  p????pte? ap? t?? ep?µe?? ???µat?sµ? ?(G)3

20
?p????e?s? ?s?µßat?? ??????t??

• ???ß??µa µ?a eta??e?a pa?a????? ??µ????
  p?????t?? ???e? ?a ta ap????e?se? se µ?a ap?????.
  ?e???? p?????ta a?t?d???? µe ??p??a ???a ?ta?
  ?????? se epaf?. ??a ? ap????? p??pe? ?a
  d?a??e?e? se pe?????? ?ste p?????ta p?? a?t?d????
  µeta?? t??? ?a µ? ß??s???ta? st?? ?d?a pe?????
• ???t?µa p???? e??a? ? e????st?? a???µ?? pe??????
  p?? apa?te?ta? ??a t?? asfa?? f??a?? t??
  p?????t??

21
? ??s?

• St?? p??a?a, ? aste??s??? d????e? ?t? t?
  a?t?st???? ?e???? ??µ???? p?????t?? p??pe? ?a µ??
  ?????? se epaf?
• St? ???f?, ?? ????f?? a?t?st?????? sta p?????ta
  ?a? e?????ta? ?ta? ta a?t?st???a p?????ta de?
  p??pe? ?a ?????? se epaf?

4 ???µata, ??at??
a b c d e f g
a - - -
b - -
c - - -
d - - -
e - - - - - -
f - - - -
g - - - -
22
???µat?sµ?? ?p?ped???? G??f??

• Te???µa ???e ep?ped?? ???f?? e??a?
  6-???µat?s?µ??
• ?p?de??? ?e epa???? st?? a???µ? t?? ????f??
  n.??µa 1. ? ?p??es? ?s??e? p??fa??? ??a n1.
  ?p?s?? ?s??e? ??a n6??µa 2. ?p???t??µe ?t? ??
  ????f?? ???? t?? s??dedeµ???? ep?ped?? ???f?? µe
  ????te?e? ap? n ????f?? µp????? ?a ???µat?st???
  µe 6 ? ????te?a ???µata.
• ??µa 3. Ta de????µe ?t? ? ???µat?sµ?? a?t?? e??a?
  ef??t?? ?a? ??a ???f??? µe n ????f??

23
S????e?a ?p?de???? (1)

• ?p?de??? (s????e?a) ?st? ?t? ? G e??a?
  s??dedeµ????, ep?ped?? ?a? ??e? n ????f??. ??te
  ap? t? ????sµa 5.6 ????µe ?t? ? G pe????e? µ?a
  ????f? v ßa?µ?? d(v)5. ?fa????µe t?? v ?a? t??
  p??s?e?µe?e? se a?t? a?µ??, ?p?te pa?????µe t?
  ???f? H.

24
S????e?a ?p?de???? (2)

• ?p?de??? (s????e?a) ? ???f?? H epe?d? ??e? n1
  ????f?? µp??e? ?a ???µat?s?e? µe 6 t? p???
  ???µata (?p??es?). ?? ?a?aß????µe st? H t??
  ????f? v ?a? t?? p??s?e?µe?e? se a?t?? a?µ??. ? v
  ??e? t? p??? 5 ?e?t?????? ????f?? ?a? ????µe st?
  d???es? µa? 6 ???µata. ?p?te ?p???e? ??a
  d?a??s?µ? ???µa ??a t?? v. ??a ? G e??a?
  6-???µat?s?µ??.

25
???µat?sµ?? ?p?ped???? G??f??

• Te???µa ???e ep?ped?? ???f?? e??a? 5-???µat?s?µ??

v1
v2
v5
v
v3
v4
Te???µa ???e ep?ped?? ???f?? e??a? 4-???µat?s?µ??
26
???µat?sµ?? ?a?t??

• G??f?? k-???µat?s?µ?? ?? p??? pe?????? (k-region
  colorable) ?? pe?????? µp????? ?a ???µat?s????
  µe k ???µata
• ???µat?sµ?? ?a?t?? ?a ß?e?e? t? e????st? p?????
  ???µ?t?? p?? apa?t???ta? ??a ?a ???µat?s?e? ??a?
  ???t??, ?ts? ?ste d?? ?e?t????? ???t? ?a µ??
  ????? t? ?d?? ???µa.
• ?? p??ß??µa ?µe??e ???t? ??a 125 ?????a ?a?
  ?µe??e ???st? ?? ? ???as?a t?? 4 ???µ?t??,
  epe?d? de? ?p???e ? s?et??? ap?de??? a? ?a? ?ta?
  eµpe????? ???st? ?t? 4 ???µata ?ta? a??et?

27
?a??de??µa

• ?st? µ?a ?pe????

28
?a??de??µa

• ???????µe ta s????a a??? ?a ???aµe ?a d?a??????µe
  t?? ???e?. ??a ???µa de? e??a? a??et?

29
?a??de??µa

• ?? p??s??s??µe ??a a??µa ???µa. Ta ?eµ????µe
  ???e ???a µe ??a ap? ta d?? ???µata.

30
?a??de??µa

• ?? p??s??s??µe ??a a??µa ???µa. Ta ?eµ????µe
  ???e ???a µe ??a ap? ta d?? ???µata.

31
?a??de??µa

• ?? p??s??s??µe ??a a??µa ???µa. Ta ?eµ????µe
  ???e ???a µe ??a ap? ta d?? ???µata.

32
?a??de??µa

• ?? p??s??s??µe ??a a??µa ???µa. Ta ?eµ????µe
  ???e ???a µe ??a ap? ta d?? ???µata.

33
?a??de??µa

• ???ß??µa ??? ?e?t?????? ???e? ?a ????? ?d??
  ???µa. ?e? fa??eta? t? s?????.

34
?a??de??µa

• ?? p??s??s??µe ??a a??µa ???µa

35
?a??de??µa

• ?e? a?????. ??e?a??µaste 4 ???µata e?a?t?a? a?t??
  t?? ???a?

36
?a??de??µa

• ?e 4 ???µata ???eta? ? ???µat?sµ??.

37
4-???µat?sµ?? ?a?t??

• ???as?a t?? 4 ???µ?t?? (4 color conjecture) ???e
  ???t?? µp??e? ?a ???µat?s?e? µe 4 ???µata
• Guthrie 1850 (pa?at???s?)
• DeMorgan 1852
• Hamilton 1852
• Cayley 1878 (de? ß???e ??s?)
• Kempe 1880 (ß???e ????? ??s?)
• Heawood 1890 (ß???e t? ????? t?? ??s??)
• Franklin 1920 (??a n25)
• Reynolds 1926 (??a n 27)
• Franklin 1931 (??a n 31)
• Winn 1943 (??a n 35)
• Ore-Stemple 1968 (??a n 40)
• Appel-Haken-Koch 1976 ß???a? µ?a ??s?
  (?p?????st???)

38
4-???µat?sµ?? ???t? ???
39
???µat?sµ?? pe??????

• Te???µa ??a? ep?ped?? ap??? ???f?? G e??a?
  k-???µat?s?µ?? (?? p??? t?? ????f??), a? ?a?
  µ???? a? ? ?e?µet????? d?ad???? ???f?? G e??a?
  k-???µat?s?µ?? ?? p??? t?? pe??????
• ????sµa ? e??as?a t?? 4 ???µ?t?? ??a t?
  ???µat?sµ? pe?????? e??a? ?s?d??aµ? µe t??
  e??as?a t?? 4 ???µ?t?? ??a t? ???µat?sµ? t??
  ????f?? ep?ped?? ???f?? ????? ß??????
• Te???µa ??a? ???t?? G e??a? 2-???µat?s?µ?? ??
  p??? t?? pe?????? a? ?a? µ???? a? e??a? Eulerian

40
???µat?sµ?? ?a?t?? vs. ????f??

• Se ???e pe????? e?s??eta? µ?a ????f?

41
???µat?sµ?? ?a?t?? vs. ????f??
Se ???e ?e???? pe?????? µe ????? s????a e?s??eta?
µ?a a?µ?
42
???µat?sµ?? ?a?t?? vs. ????f??
? ???µat?sµ?? t?? pe?????? e??a? ?s?d??aµ?? µe t?
???µat?sµ? t?? d?ad???? ???f??
43
?s??s? 3

• ?a ???µat?s?e? ? ep?µe??? ???t?? µe 4 ???µata

44
??s?

• Ta µp????saµe ?a a?apa?ast?s??µe t? ???t? µe ??a
  ???f? ?p?? ?? ????f?? a?t?st?????? st?? ???e? ?a?
  ?? a?µ?? e?????? ?e?t?????? ???e?.
• ? ???µat?sµ?? t?? ????f?? t?? ???f?? a?t?st???e?
  st? ???µat?sµ? t?? pe?????? t?? ???t?.

45
???µat?sµ?? ??µ??

• G??f?? k-???µat?s?µ?? ?? p??? a?µ??, k-edge
  colorable ?? a?µ?? µp????? ?a ???µat?s???? µe
  k ???µata, ?ste 2 a?µ?? p??sp?pt??se? st?? ?d?a
  ????f? ?a ????? d?af??et??? ???µa
• G??f?? k-???µat???? ?? p??? a?µ??, k-edge
  chromatic ?? a?µ?? µp????? ?a ???µat?st??? µe k
  ???µata, a??? ??? µe k1
• ?? k ???eta? ???µat???? a???µ?? a?µ?? -edge
  chromatic number- ? ???µat???? ?at??????
  -chromatic index- ?a? s?µß????eta? µe ?'(G)k

46
???µat?sµ?? ??µ??

• ?st? ??a? ???f?? G µe µ???st? ßa?µ? ????f?? D(G).
  ???a? p??fa??? ?t? ?'(G)D(G)
• Te???µa (Vizing 1964) ?? G e??a? ??a? ap???
  ???f??, t?te ?s??e? D(G)?'(G)D(G)1
• ???a? ???t? p??ß??µa ? p??sd????sµ?? t?? ???f??
  µe ???µat??? a???µ? a?µ?? D(G) ?a? D(G)1

47
??d???? ?e??pt?se??

• ???a? d??at? ?a ß?e?e? ? ???µat???? a???µ?? a?µ??
  se e?d???? pe??pt?se?? ???f??. G?a pa??de??µa
• ??(C2n)2
• ??(C2n1)3
• ??(Wn)n1, a? n4

48
??µe?e?? G??f??

• Te???µa Koenig ?st? G ??a? d?µe??? ???f?? µe
  µ???st? ßa?µ? ????f?? D. ??te ?s??e? ??(G) D.
• ???? d?at?p?s?
• Te???µa G?a ???e p???? d?µe?? ???f? ?s??e?
  ??(Km,n)D(Km,n)max(m,n)

49
?p?de??? ??a ????? ??µe?? G??f?

• ?p?de??? ?st? ?t? mn. ?st? ep?s?? ?t? ? ???f??
  pa??st??eta? ??af??? ?p?? st? p??????µe?? s??µa
  d??. ?? m ????f?? ß??s???ta? ep? µ?a? e??e?a?
  ??aµµ??, e?? ?? n ????f?? t?p??et???ta?
  ?aµ???te?a ep? µ?a? ????? e??e?a?. ? ???µat?sµ??
  t?? ???f?? ep?t?????eta? ???µat????ta? d?ad?????
  t?? a?µ?? p?? p??s?e??ta? st?? n ????f?? ?at? t??
  ???????a?? f??? ?a? ???s?µ?p????ta? ta ???µata
  1,2,,m, 2,3,,m,1,,n,,m,1,..,n-1.

50
????e?? G??f??

• Te???µa G?a ???e p???? ???f? Kn ?s??e?
• ??(Kn)

51
?p?de??? (pe??tt? t???)

• Te???µa G?a ???e p???? ???f? Kn ?s??e?
• ??(Kn)
• ?p?de??? ?pe?d? ???e ????f? ??e? ßa?µ? n1 ap?
  t? ?e???µa t?? Vizing ????µe ?t? ?'(G)n ?
  ?'(G)n?1. ?? t? n e??a? pe??tt??, t?te t? p?????
  t?? a?µ?? µe ?d?? ???µa e??a? t? p??? (n-1)/2
  (p.?. K3 ? ?5), d??t? d?af??et??? 2 ap? a?t?? t??
  a?µ?? (?d??? ???µat??) s??a?t??ta? se µ?a ?????
  ????f?. ???? ? p????? ???f?? Kn ??e? a???ß??
  n(n-1)/2 a?µ??, ??a ? a???µ?? t?? d?af??et????
  ???µ?t?? t?? a?µ?? t?? p??pe? ?a e??a? t??????st?
  n. ??a ?'(G)n.

52
?p?de??? (??t?a t???)

• Te???µa G?a ???e p???? ???f? Kn ?s??e?
• ??(Kn)
• ?p?de??? ?? t? n e??a? ??t??? ?a de????µe ?t?
  ??(Kn)n1, d????ta? ??a (n1) ???µat?sµ? t??
  a?µ?? t?? Kn. ?? n2, t?te p??fa??? ??e?a??µaste
  1 ???µa. ?? ngt2, d?a???f??µe µ?a ????f? v µa?? µe
  ??e? t?? p??s?e?µe?e? a?µ??. ??te p????pte? ?
  ???f?? Kn-1 p?? ??e? pe??tt? p????? a?µ?? ?a?
  s??ep?? e??a? n-1 ???µat???? ?? p??? t?? a?µ??.
  ?p? ???e ????f? ?e?pe? a???ß?? ??a ???µa ?a? ??a
  a?t? ta ???µata e??a? d?af??et??? µeta?? t???.
  S??ep??, ?? a?µ?? t?? Kn p?? p??s?e??ta? st??
  ????f? v µp????? ?a ???µat?s???? µe ta ???µata
  p?? ?e?p???. S??ep?? ??(Kn)n1, a? n e??a?
  ??t???.

53
???µat?sµ?? ??µ?? - ???????f??

• ?? ????µe p??????f? (µe pa??????e? a?µ??), t?te
  ??????µe ?? µ???st? p???ap??t?ta -maximum
  multiplicity, m(G) t? µ???st? p????? a?µ?? p??
  e?????? 2 ?p??esd?p?te ????f??.
• Te???µa (Vizing Extended Version, 1965) ?? G
  e??a? p??????f??, t?te D(G)??(G)D(G)m(G)
• Te???µa (Tait, 1880) ??a? ???t?? G e??a?
  4-???µat?s?µ?? ?? p??? t?? pe?????? a? ?a? µ???
  a? e??a? 3-???µat?s?µ?? ?? p??? t?? a?µ??
• Te???µa Shannon ?st? G ??a ???f?? (µp??e? ?a
  e??a? ?a? p??????f??) µe µ???st? ßa?µ? ????f?? D.
  ??te ?s??e?
• D ??(G) 3D/2, a? D ??t???
• D ??(G) (3D-1)/2, a? D pe??tt??

54
F???µata ???µat???? ?ata?????

• G?a ?a ß???µe t? ???µat??? ?at????? ??(G) e???
  ???f?? G ????? ß?????? ß??s???µe ??a ep??? ?a?
  ??a ??t? ????. ?? a?t? ta 2 ???a e??a? ?sa t?te
  t? ??(G) ?s??ta? µe a?t? t?? t?µ?.
• ???a?? ep??? ???a ??a t? ??(G)
• ?? µ??e??? m t?? G
• D1, ap? ?e???µa Vizing (a? t? G e??a? ap??)
• Dm(G), ap? ?e???µa Vizing extended (a? t? G
  e??a? p??????f??), m(G) e??a? t? p????? t?? a?µ??
  p?? s??d???? 2 ?p??esd?p?te ????f??.
• 3D/2 ? (3D1)/2 ap? ?e???µa Shannon
• ???a?? ??t? f???µata ??a t? ??(G)
• D, ? µ???st?? ßa?µ?? t?? ????f?? t?? G

55
?s??s? 4

• ???t?s? ?? s?µpe?a????µe ??a t??? ???f??? G ??a
  t??? ?p????? ?s??e? ?'(G)2
• ??s? ?? ?'(G)2, t?te p???e?ta? ??a ???f???
  ap?te???µe???? ap? s???st?se? p?? e??a? e?te
  ?????? ??t??? µ?????, e?te ap?µ???µ??e? ????f??,
  e?te µ???p?t?a (Pn, n3). ?p?s?? µ?a t??????st?
  s???st?sa e??a? ?????? ??t??? µ????? ? ???f??
  µ???p?t? (Pn, n3).

56
?s??s? 5

• ???t?s? ????? ? ???µat???? ?at?????? t?? K2,3
• ?pa?t?s? ?'(K2,3) 3, Te???µa 6.12

57
?s??s? 6

• ???t?s? G?a t?? p???? d?µe?? ???f? K2,4 ?a
  ?p?????st??? ta ep??? ?a? ta ??t? ???a t??
  ???µat???? ?ata????? ?'(G) s?µf??a µe t? Te???µa
  t?? Vizing, ? p?a?µat??? t?µ? ?'(G) ?a? ??a?
  ???µat?sµ?? a?µ?? µe ?'(G) ???µata.
• ?p??t?s? 4?'(G)5. ??a?µat??? t?µ? ?'(G)4
  (Te???µa Koenig ? Te???µa 6.12).

58
?s??s? ???µat?sµ?? ?a??d???

• ??a? µ??a????? ???e? ?a ?atas?e??se? ??a taµp??
  ape?????s?? (ep?de????) µ?sa st? ?p??? p???e?ta?
  ?a t?p??et????? ?a? ?a d?as??de???? ??e?t????
  st???e?a a,b,.?a ?a??d?a p?? p???e?ta? ?a
  s??de???? st? st???e?? a ß?a????? ap? µ?a ?p? t??
  taµp??, ta ?a??d?a p?? p???e?ta? ?a s??de???? st?
  st???e?? b ß?a????? ap? µ?a ???? ?p? t?? taµp??
  ?.?.?. G?a ?a d?a??????ta? ta ?a??d?a p??
  ß?a????? µ?sa ap? t?? ?d?a ?p? p??pe? ?a ?????
  d?af??et??? ???µa. ????? e??a? ? e????st??
  a???µ?? ???µ?t?? p?? apa?t???ta?

59
??s?

• ??apa??st??µe ta s?µe?a d?as??des?? µe t??
  ????f?? e??? ???f??, e?? ta ?a??d?a µe t?? a?µ??
  t??. ? d?p?a??? ???f?? a?apa??st? ??a taµp?? 6
  st???e???. ?pe?d? ? ????f? b ??e? 5 p??sp?pt??se?
  a?µ?? ?a? p??pe? ??e? ?? a?µ?? ?a ???µat?s????
  d?af??et??? s?µpe?a????µe ?t? apa?t???ta?
  t??????st?? 5 ???µata. St?? p???? 5 ???µata e??a?
  a??et?, ?p?? fa??eta? st? d?p?a?? ???µat?sµ?

60
???µat??? ???????µa

• ???t?s? ?at? p?s??? d?af??et????? t??p???
  µp????µe ?a ???µat?s??µe t?? ????f?? e??? ???f??
  µe k ???µata
• ?p??t?s? (Birkhoff 1912) ???µat??? p??????µ?
  chromatic polynomial ? ???µat??? s????t?s?
  chromatic function, ???µ??eta? ? a???µ?? t??
  t??p?? p?? µp????? ?a ???µat?s???? ?? ????f??
  e??? ???f?? G µe k ???µata ?a? s?µß????eta? µe
  PG(k).

61
?a??de??µa

• ? ???µat??? s????t?s? t?? ep?µe??? ???f?? e??a?
  PG(k)k(k-1)2.
• ?p?de??? ?st? ?t? ? µesa?a ????f? µp??e? ?a
  ???µat?ste? µe ?p???d?p?te ap? ta k ???µata. ??te
  ?? 2 a???a??? ????f?? µp????? ?a ???µat?st??? µe
  ?p???d?p?te ap? ta ap?µ????ta k-1 ???µata.

62
S??se??

• ?a ap?de?????? ?? s??se??
• PNn(k)kn
• PT(k) k(k1)n-1, a? T e??a? d??t??
• PKn(k)k(k1)(k2)(kn1)
• PG(k)0 a? klt?(G)
• PG(k)gt0 a? k?(G)

63
?p?de??e??

• PNn(k)kn, ??at? ???e ????f? t?? µ?de????? ???f??
  µp??e? ?a ???µat?s?e? µe ?p???d?p?te ap? ta
  d?a??s?µa k ???µata.
• PT(k) k(k1)n-1, a? T e??a? ??a d??t?? n
  ????f??. ?e?????µe ap? ??a f???? (? ???a) p??
  µp??e? ?a ???µat?s?e? µe ?p???d?p?te ap? ta k
  ???µata. ???a????µe se µ?a ?e?t????? µ?
  ???µat?sµ??? ????f? t?? f?????, ? ?p??a µp??e? ?a
  ???µat?s?e? µe ?p???d?p?te ap? ta ?p????pa k-1
  ???µata. ???a????µe se µ?a ?e?t????? µ?
  ???µat?sµ??? ????f? t?? p??????µe??? ????f?? ?
  ?p??a µp??e? ?a ???µat?s?e? µe ?p???d?p?te ap? ta
  ?p????pa k1 ???µata ?.?.?.

64
?p?de??e??

• PKn(k)k(k1)(k2)(kn1), e??a? p??fa??? d??t?
  st?? p???? ???f? ?n ?p?? ??e? ?? ????f?? e??a?
  ?e?t?????? µeta?? t???, a? ? 1? ????f?
  ???µat?s?e? µe ?p???d?p?te ap? ta k d?a??s?µa
  ???µata, ? 2? µp??e? ?a ???µat?s?e? µe
  ?p???d?p?te ap? ta k1 d?a??s?µa ???µata, ..., ?
  n-?st? µe ?p???d?p?te ap? ta k (n1)kn1
  d?a??s?µa ???µata

65
?p?de??e??

• PG(k)0 a? klt?(G), d??t? ? ???µat???? a???µ??
  ?(G) e??a? ? e????st?? a???µ?? ???µ?t?? p??
  apa?t???ta? ??a t?? ?at?????? ???µat?sµ? t??
  ????f?? e??? ???f?? G. ??a a? ????µe ????te?a
  ???µata st? d???es? µa? de? ?p???e? t??p?? ?a
  ???µat?s?e? ? ???f??.
• PG(k)gt0 a? k?(G), d??t? a? ????µe pe??ss?te?a
  ???µata ap? t? ???µat??? a???µ? ?(G), t?te
  s?????a ?a ?p?????? ??p???? t??p?? ???µat?sµ??
  e??? ???f?? G.

66
???as?a 4 ???µ?t??

• ? e??as?a t?? 4 ???µ?t?? µp??e? ?a e?f?as?e? ??
  e??? a? G e??a? ??a? ap??? ep?ped?? ???f??, t?te
  PG(4)gt0.
• G?a ??a t??a?? ???f? G de? e??a? e????? ?p??es? ?
  e??es? t?? ???µat???? p??????µ??. ?? ep?µe??
  ?e???µa ß???? p??? a?t? t?? ?ate????s?.

67
Te???µa ???µat???? ???????µ??

• Te???µa ?st? ?t? v ?a? w e??a? µ? ?e?t??????
  ????f?? e??? ap??? ???f?? G. ?? G1G(v,w) ?a?
  G2G/(v,w) t?te ?s??e?
• PG(k)PG1(k)PG2(k)
• ? p???? ???eta? s??des? (join) ? ?????sµa
  (sum).
• ? p???? / ???eta? s?????e?s? (fusion, merge).
• ? e??s?s? µp??e? ?a ??afe? ?a? ?? e???
• PG1(k) PG(k) PG2(k)

68
?fa?µ???

• ?fa?µ??? t?? ?e???µat?? st? ep?µe?? s??µa d??e?
  t? s??s? k(k1)(k2)2 k(k1)(k2)(k3)k(k1)(
  k2)

69
????sµa

• ? ???µat??? s????t?s? e??? ap??? ???f?? G µe n
  ????f?? e??a? ??a p??????µ? ßa?µ?? n ?? p??? k.
  ?p?s?? t? p??????µ? a?t? ??e? a???a????
  s??te?est?? µe e?a??ass?µe?a p??s?µa, ??
  µe?a??te?? ??? t? kn ?a? ?? sta?e?? ??? t? 0.

70
?a??de??µa

• ?? efa?µ?s??µe t? p??????µe?? ?e???µa st??
  ep?µe?? ???f? G

71
?a??de??µa
72
?a??de??µa

• G ?5 3?4 2?3
• ??a
• PG(k) PK5(k) 3PK4(k) 2PK3(k)
• k(k-1)(k-2)(k-3)(k-4) 3k(k-1)(k-2)(k-3)
  2k(k-1)(k-2) k5 7k4 19k3 23k2 10k

73
?a??de??µa µe t?? ???? s??s?
G1
G
G2
-

PG(k) k4 3k3 3k2 k
74
???se???st???? ???????µ?? ???µat?sµ??

• ? p??sd????sµ?? t?? ???µat???? a???µ?? e??a?
  d?s?e???st? (intractable) p??ß??µa. ??t??a
  p??ß??µata e??a? e?e??a p?? de? ?????ta? se
  p??????µ??? ?????
• ? ???? e???st???? ???s?µ?p??e?ta? ??a a??????µ???
  p?? ß??s???? ??se?? se ??p??? p??ß??µa a??? de?
  ?p???e? e????s? ?t? a?t? ? ??s? e??a? ? ?a??te??.
  ?p????? ?a ?e??????? ?t? e??a? p??se???st???? ?a?
  ??? epa???ße?? a??????µ??

75
Se???a??? ???????µ?? (?p??st??)

• ??e? ?? ????f?? a???µ???ta? ap? v1 ??? vn. ??a ta
  ???µata a???µ???ta? ap? 1 ??? n. H ?d?a
  s???stata? st? ???µat?sµ? t?? ????f?? µ?a p???
  µ?a a?????a µe t? se??? e??tas?? t??? ?a?
  ???s?µ?p????ta? t? µ????te?? ???µa p?? e??a?
  d?a??s?µ?, d??ad? de? ??e? ???s?µ?p????e? se
  ?e?t?????? ????f??.

76
? ???????µ??
77
?a??de??µa

• ?a ???µat?s?e? ? ep?µe??? ???f?? µe t? se???a??
  µ???d? a? ?p?te?e? ?t? ? ??sta t??a?a?
  ta????µ?s?? t?? ????f?? t?? e??a? L
  v1,v2,v4,v3,v5,v6,v7

• Se??? ???µ?t??
• ???????
• ??t????
• ?p?e
• ???s???
• ?
• ?
• ?

v3
v4
v2
v7
v1
v5
v6
78
?a?a?t???st???

• ? se???a??? a??????µ?? ??e? p???p????t?ta ?(nm)
• ?? ?? ????f?? e?etas???? µe d?af??et??? se???
  µp??e? ?a p???????? d?af??et???? ???µat????
  t??e??
• ?? p??spa??s??µe ?a ???µat?s??µe ??a d?µe?? ???f?
  n ????f?? ?aµß????ta? t?? ????f?? e?a???? ap? ta
  2 s????a ????f??, t?te ?a ??e?as???µe n/2 ???µata
  a?t? ??a 2 p?? a?t? ?a µp????se ?a ???µat?s?e?

79
?p?p???? ???et???? ????d?? ???ta ? ?e?a??te??

• ???ta ? µe?a??te?? (largest first) ?? ????f??
  ta????µ???ta? ?at? f?????sa t??? a?????a µe t?
  ßa?µ? t???. Se pe??pt?s? ?s?pa??a? se s??s? µe
  t??? ßa?µ??? t?? ????f?? ep????eta? t??a?a µ?a
  ?p???f?a ????f?. ?at?p?? a??????e?ta? ? se???a??
  µ???d??.

80
?fa?µ???

• ?a ???µat?ste? t?? ep?µe?? ???f? µe t? µ???d?
  p??ta ? µe?a??te??

• Se??? ???µ?t??
• ???????
• ??t????
• ?p?e
• ???s???
• ?
• ?
• ?

v3
v4
v2
v7
v1
v5
v6
81
?p?p???? ???et???? ????d?? ?e?e?ta?a ? ?????te??

• ?e?e?ta?a ? µ????te?? (smallest last) a????? ??
  ????f?? ta????µ???ta? ?at? f?????sa se??? t??
  ßa?µ?? t???. Se pe??pt?s? ?s?pa??a? ?? p??? t???
  ßa?µ??? t?? ????f?? ep????eta? t??a?a µ?a
  ?p???f?a ????f?. St? s????e?a ? ????f? µe t?
  µ????te?? ßa?µ? d?a???feta? ap? t? ???f? µa?? µe
  t?? p??sp?pt??se? a?µ?? t?? µe s??p? ?a
  ???µat?s?e? te?e?ta?a. ?p?s?? e??µe?????ta? ??
  ßa?µ?? t?? ?p????p?? ????f?? ?a? d?a???feta? p???
  ? ????f? µe t? µ????te?? ßa?µ? ? ?p??a
  ???µat??eta? p??te?e?ta?a ???. ?ts? ???eta? µ?a
  ta????µ?s? t?? ????f?? ?a? a??????e? ? se???a???
  ???µat?sµ?? t?? ???f??.

82
?fa?µ???

• ?a ???µat?ste? ? ep?µe??? ???f?? µe t? µ???d?
  te?e?ta?a ? µ????te??. ?? ????f?? ?atat?ss??ta?
  a????? ?at? f?????sa t??? ßa?µ?? ?? e???
  v1,v2,v3,v4,v5,v6,v7. ??a ?at?ta?? t?? ????f??
  e??a? v1,v5,v6,v4,v3,v2,v7. ?fa?µ????ta? t?
  se???a?? ???µat?sµ? st?? p??????µe?? se???
  pa?????µe

• Se??? ???µ?t??
• ???????
• ??t????
• ?p?e
• ???s???
• ?
• ?
• ?

v3
v4
v2
v7
v1
v5
v6
83
?p?p???? ???et???? ????d?? ?a?µ?? ???µat??

• ? µ???d?? t?? ßa?µ?? ???µat?? (Brelaz) ?a?µ??
  ???µat?? (color degree) µ?a? ????f?? v ????eta?
  ?? ? a???µ?? t?? ???µ?t?? p?? ???s?µ?p??????a?
  ??a t? ???µat?sµ? t?? ?e?t?????? ????f?? t?? v.
  (?p???e? ?a??te?a pe??pt?se?? ?s?pa??a? ?? p???
  ßa?µ??? ????f??)

84
???????µ?? ?a?µ?? ???µat??
85
?a??de??µa

• ?a ???µat?ste? ? ep?µe??? ???f?? µe t? µ???d?
  Brelaz

86
?a??de??µa

• ?? ????f?? ?atat?ss??ta? a????? ?at? f?????sa
  t??? ßa?µ?? (µp?e ???µa) ?? e??? v1(4), v2(4),
  v3(3), v4(3), v5(3), v6(3), v7(2). ? v1
  ???µat??eta? µe t? ???µa 1 (??????? ???µa). ??
  ?e?t?????? ????f?? t?? v1 d??ad? ?? v2,v5, v6,
  v7 ap??t??? ??e? ßa?µ? ???µat?? 1 (p??s???
  ???µa).

87
?a??de??µa

• ?eta?? a?t?? ep????eta? p??? ???µat?sµ? ? ????f?
  v2 epe?d? ??e? t? µ???st? ßa?µ? (4) ?a?
  ???µat??eta? µe t? ???µa 2. ?a??????a
  e??µe?????ta? ?? ßa?µ?? ???µat?? t?? µ?
  ???µat?sµ???? ?e?t?????? ????f??. St? s????e?a
  ???µat??eta? ? v7 (d??t? ??e? µ???st? ßa?µ?
  ???µat??) µe t? ???µa 3.

88
?a??de??µa

• St? s????e?a ?p???f?e? e??a? ??e? ?? ????f?? af??
  ????? ?d?? ßa?µ? ???µat?? (1) ?a??? ?a? ?d??
  ßa?µ? (3), ?p?te ep????eta? t??a?a ? v3 ? ?p??a
  ???µat??eta? µe t? ???µa 1. St? s????e?a ? v4
  ???µat??eta? µe t? ???µa 3.

89
?a??de??µa

• St? s????e?a ???µat??eta? ? v5 µe t? ???µa 2 ?a?
  t???? ? v6 ???µat??eta? µe t? ???µa 3

90
?a????sµata

• ??a ta???asµa (matching) e??? ???f?? G e??a? ??a
  s????? a?µ?? t?? ?p?? de? ?p?????? 2 a?µ?? p?? ?a
  p??sp?pt??? st?? ?d?a ????f?.
• ???e ???f?? µp??e? ?a d?aspas?e? se ta????sµata
  d??t? st?? ap?? pe??pt?s? a? ?p?????? m a?µ??
  µp????? ?a s??µat?s???? m ta????sµata µe µ?a a?µ?
  t? ?a???a.
• Se µ?a d??spas? a?µ?? e??? ???f??, ???e a?µ? t??
  a???e? se ??a µ??? ta???asµa
• ?? p??ß??µa t?? p??sd????sµ?? t?? e????st??
  a???µ?? ta???asµ?t?? ??a t? d??spas? e??? ???f??
  e??a? d?sep???t?.

91
?a??de??µa

• S?µf??a µe t?? ep?µe?? ???µat?sµ? a?µ?? ?? a?µ??
  t?? d?asp??ta? se 5 s????a ???a µeta?? t???. ??
  a?µ?? t?? ?d??? s?????? ???µat????ta? µe t? ?d??
  ???µa ?a? ap?te???? ??a ta???asµa.
• af, bc, ab, cd, ef, ab, de, ab, cf, de,
  bc, de

92
?? ???ß??µa t?? ??et?se??

• St? t???? t?? a?ad?µa???? ?t??? ???? ?? f??t?t??
  p??pe? ?a d?s??? e?et?se?? µ?a? ??a? µe ??p?????
  ap? t??? ?a????t?? t???. ??se? e?etast????
  pe???d?? ??e?????ta?
• ?? ?e???s??µe t? ap?? pa??de??µa p?? ?p?????? 4
  f??t?t?? a, b, c, d ?a? 3 ?a????t?? A, B, C.
• ??apa??st??µe t??? f??t?t?? ?a? t??? ?a????t?? µe
  t?? ????f?? e??? d?µe???? ???f?? ?a? e?????µe µ?a
  ????f?-f??t?t? µe µ?a ????f?-?a????t? ?ta? ?
  s???e???µ???? f??t?t?? p??pe? ?a e?etas?e? ap? t?
  s???e???µ??? ?a????t?.

93
?? ???ß??µa t?? ??et?se??

• ?? 2 a?µ?? p??sp?pt??? st?? ?d?a ????f?, t?te ??
  a?t?st???e? e?et?se?? de? µp????? ?a
  p?a?µat?p??????? ta?t?????a. S??ep?? t? p??ß??µa
  a???eta? se p??ß??µa d??spas?? se s????a a?µ??,
  sta ?p??a ?a µ?? ?p???e? ?e???? a?µ?? p?? ?a
  s??a?t??ta? st?? ?d?a ????f?. ???ad? ?????µe
  d??spas? se ta????sµata.

94
?? ???ß??µa t?? ??et?se??

• St?? p???e?µ??? pe??pt?s? ? e????st?? a???µ??
  ta???asµ?t?? e??a? 3 ?a? ??a ?at??????
  ?????-d????aµµa t?? e?et?se?? fa??eta? st?
  s????e?a.
• ?a s????a a?µ?? e??a? aA, bB, dC, aC, bA,
  cB, bC, cA, dB

95
?? ???ß??µa t?? ??et?se??

• ?? ?d?? p??ß??µa µp??e? ?a ?e????e? ?a? p??ß??µa
  ???µat?sµ?? a?µ??. ?? ???µat?s??µe t?? a?µ?? t??
  p??t?? s?????? ??????e?, t?? de?t???? ??t???e?
  ?a? t?? t??t?? s?????? µp?e t?te ta ???µata se
  ???e ????f? t?? ???f?? (e?te f??t?t? e?te
  ?a????t?) e??a? d?af??et??? µeta?? t???. ??
  a?µ?? µe t? ?d?? ???µa ap?te???? ??a ta???asµa.
  G?a t?? ep?µe?? ???f? ?'(G)3.

96
???? ???ß??µa ??et?se??

• ?st? ?t? p??pe? ?a e?d??e? p????aµµa e?et?se??
  ??a ta µa??µata µe t??? e??? ??d?????
• 1007, 3137, 3157, 3203, 3261, 4115, 4118, 4156
• Se ??p??a ?e??? µa??µ?t?? de? ?p?????? ??????
  f??t?t??
• 1007-3137, 1007-3157, 3137-3157, 1007-3203,
  1007-3261, 3137-3261, 3203-3261, 1007-4115,
  3137-4115, 3203-4115, 3261-4115, 1007-4118,
  3137-4118, 1007-4156, 3137-4156, 3157-4156
• ????? e??a? ? e????st?? a???µ?? e?etast???? slot?

97
???? ???ß??µa ??et?se??

• ?? p??ß??µa ?a ep????e? µe ???µat?sµ? a?µ??. ??
  ????f?? e??a? µa??µata, ?? a?µ?? d???????
  e?et?se?? p?? de? µp????? ?a d?e?a?????
  ta?t?????a.

98
???? ???ß??µa ??et?se??

• ?p????????µe t? s?µp????µat??? ???f?

3203
3261
3137
4115
1007
4118
3157
4156
99
???? ???ß??µa ??et?se??
?p????????µe t? s?µp????µat??? ???f?
3203
3261
3137
4115
1007
4118
3157
4156
100
???? ???ß??µa ??et?se??
?a?as?ed?????µe
3137
3203
3261
4115
1007
3157
4118
4156
101
???? ???ß??µa ??et?se??
? ???f?? de? e??a? 1-???µat?s?µ?? ???? µ?a?
?p??asd?p?te a?µ?? (C2)
3137
3203
3261
4115
1007
3157
4118
4156
102
???? ???ß??µa ??et?se??
? ???f?? de? e??a? 2-???µat?s?µ?? ???? t?? C3
3137
3203
3261
4115
1007
3157
4118
4156
103
???? ???ß??µa ??et?se??

• ???µat????µe µe 3 ???µata ???????, µp?e, p??s???

3137
3203
3261
4115
1007
3157
4118
4156
104
???? ???ß??µa ??et?se??

• ?st? 3203-???????, 3157-µp?e, 4118-p??s???

3137
3203
3261
4115
1007
3157
4118
4156
105
???? ???ß??µa ??et?se??

• ??a 4156-µp?e

3137
3203
3261
4115
1007
3157
4118
4156
106
???? ???ß??µa ??et?se??

• ??a 3261 ?a? 4115 ep?s?? ??????a

3137
3203
3261
4115
1007
3157
4118
4156
107
???? ???ß??µa ??et?se??

• ?a 3137 ?a? 1007 e??a? e????? pe??pt?s?

3137
3203
3261
4115
1007
3157
4118
4156
108
???? ???ß??µa ??et?se??

• ??a apa?t???ta? 3 e?etast??? slot.

Slot 2
3137
3203
3261
Slot 1
4115
1007
3157
4118
Slot 3
4156
109
???ß??µata ?????p????aµµat?sµ??

• Se pa??µ??a p??ß??µata ?????p????aµµat?sµ?? ??
  a?t?st????? ???f?? e??a? d?µe?e??. S??ep?? ta
  p??ß??µata a?t? a?????ta? st?? e??es? t??
  ???µat???? a???µ?? a?µ?? e??? d?µe???? ???f??.
• Se a?t? t? p??ß??µa d??eta? ap??t?s? ap? t?
  Te???µa t?? Koenig (?'(G)D).
• ? e????st?? a???µ?? ta???asµ?t?? ?s??ta? µe t?
  ???µat??? a???µ? a?µ?? t?? d?µe???? ???f??, ?
  ?p???? µe t? se??? t?? ?s??ta? µe t? µ???st?
  ßa?µ? ????f?? t?? ???f??.