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五色定理の証明
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pazzle1230
September 03, 2018
Science
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五色定理の証明
pazzle1230
September 03, 2018
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Transcript
ޒ৭ఆཧͷূ໌ άϥϑཧ @pazzle1230
ޒ৭ఆཧͱ ▸ ਤΛͲͷ2৭ྡΓ߹Θͳ͍Α͏ʹ5৭ͰృΓ͚ΒΕΔ ▸ ฏ໘తάϥϑʹ͓͍ͯΛ5৭ͰృΓ͚ΒΕΔ ▸ ΑΓڧ͍࢛৭ఆཧ͕͋Δ͕͍ͨ͠Ί͍͠
༻ޠ ▸ ฏ໘తάϥϑ ▸ ล͕Ҏ֎ͰަΘΒͳ͍Α͏ʹͰ͖Δ ▸ ɹɹ࠼৭ ▸ ৭ͰృΓ͚ΒΕΔͱ͖ͷృΓ͚ํ ▸
ࣸ૾ Ͱ͋Γɼล ͕͋Δͱ͖ k− k c : V → {1,⋯, k} uv c(u) ≠ c(v)
༻ޠ ▸ છ৭: ▸ ࠼৭͕ଘࡏ͢Δ࠷খͷ ▸ ಉ৭ू߹ ▸ ಉ৭ΛృΒΕͨͷू߹( ݸ͋ΓɼͦΕͧΕಠཱू߹)
▸ ฏۉ࣍ k− k k χ(G) d(G)
༻ޠ ▸ Ϧετ࠼৭ ▸ ɹɹɹ ͕༩͑ΒΕͨͱ͖ͯ͢ͷ ʹରͯ͠ ▸ બ(Ϧετ࠼৭): ͱද͢
▸ Ͱ͋Δάϥϑͯ͢ͰϦετ࠼৭Մೳͳ࠷খͷ ▸ ͯ͢ͷϦετ͕ Ͱ͋Δͱ͖ී௨ͷ࠼৭ͳͷͰڧ͍ ▸ ͯ͢ͷάϥϑͰ (Sv )v∈V v ∈ V c(v) ∈ Sv |Sv| = k k ch(G) {1,⋯, k} ch(G) ≥ χ(G)
Ϧετ࠼৭ͷྫ
Ϧετ࠼৭ΛΘͳ͍ূ໌ ▸ ͬͪ͜ϝΠϯ͡Όͳ͍ͨΊ͓ؾ࣋ͪײΛߴ͍ͯ͘͘͠ ▸ ฏ໘ࡾ֯ܗׂ͠EulerͷެࣜͰؤுΔͱ ͕Ͷ ▸ ࣍5ҎԼͷ ͕ଘࡏ͢Δ ▸
nͷؼೲ๏Ͱূ໌͢ΔͷͰ ࣗ໌ʹཱɼ ͰΔ m ≤ 3n − 6 d(G) = 2m/n ≤ 2(3n − 6)/n < 6 n ≤ 5 n > 5 v
͓ؾ࣋ͪূ໌(ূ໌Ͱͳͦ͞͏) Ϧετ࠼৭ΛΘͳ͍ূ໌ ▸ ؼೲ๏ͷԾఆΑΓ 5-࠼৭Մೳ ▸ ʹྡ͢ΔͰ5৭ΘΕͯΔͱԾఆ͢Δ ▸ 4৭͔͠ΘΕ͍ͯͳ͚ΕΓͰృͬͯOK H
:= G − v v
͓ؾ࣋ͪূ໌(ূ໌Ͱͳͦ͞͏) Ϧετ࠼৭ΛΘͳ͍ূ໌ ▸ 5ͭͷྡΛ࣌ܭճΓʹ ͱ͠ɼ৭ ▸ ͕1ͱ3ͷͷΈ͔ΒͳΔύεͰܨ͕ΕͯΔͱԾఆ ▸ ඇ࿈݁ͳΒ Λ
ͱಉ৭ʹؤுͬͯͰ͖͖ͯͬ͞ͱҰॹ ▸ ͷ࿈݁Ͱ1ͱ3Λަͨ͠Β ͷ৭มΘΒͳ͍ͨΊ v1 , ⋯, v5 v1 , v3 v3 v1 v1 1,⋯,5 v3
͓ؾ࣋ͪূ໌(ূ໌Ͱͳͦ͞͏) Ϧετ࠼৭ΛΘͳ͍ূ໌ ▸ ܨ͕ͬͯͨΒฏ໘άϥϑͳͷͰɹɹɹ͕அ͞ΕΔ ▸ ͷύε1ͱ3ͷͷΈͰ ύεަΘΒͳ͍ ▸ Λಉ͡৭ʹͰ͖Δ ▸
݁ہ ʹྡ͢Δ4৭ҎԼʹͳΓ5৭ͰృΕΔ v2,v4 v1,v3 v2,v4 v2,v4 v
Ϧετ࠼৭ʹΑΔূ໌ ▸ ࣌ؒతʹͬͺΓ͓ؾ࣋ͪ ▸ ࣍ͷओுΛূ໌͢Δ ▸ خ͍͠ཧ༝ ▸ ͷۃେฏ໘άϥϑΛߟ͑Δͱ֎ྖҬΛؚΉશ͕ͯࡾ֯ܗͰׂ ▸
֎ྖҬͱͷڥքʹؚ·ΕΔ3ͷ͏ͪɼ2ͭΛృΓ Λద༻Ͱ͖Δ શͯͷྖҬ͕ࡾ֯ܗͰɼ֎ྖҬͱͷڥքʹؚ·ΕΔด࿏ ͕͋ΔͱԾఆ͢Δɽ ʹͦΕͧΕ1,2ΛృΓɼ ͷ֤ʹ3৭Ҏ্ͷ Ϧετɼ ͷ֤ʹ5৭Ҏ্ͷϦετ͕༩͑ΒΕ͍ͯΔͱ͖ɼ ͷ࠼৭ Λ֦ு͠ ͷ࠼৭͕ߏͰ͖Δ C = v1 ⋯vk v1 v1 , v2 C G − C v1 , v2 G G ( * ) ( * )
Ϧετ࠼৭ʹΑΔূ໌ ͷओு ( * ) (࣮ࡍʹͦΕͧΕ3৭Ҏ্ɼ5৭Ҏ্Ͱ৭ҙ͕ͩɼ؆୯ͷͨΊ1ʙ5ͷΈ༻)
Ϧετ࠼৭ʹΑΔূ໌ ▸ ʹؔ͢Δؼೲ๏Ͱূ໌͢Δ ▸ ͷͱ͖ɼ ͳͷͰࣗ໌ ▸ ͱ͕ͯ͠ΑΓগͳ͍άϥϑͰཱ͢ΔͱԾఆ ▸ ͜͜Ͱ
͕ݭ Λͭ߹ͱ࣋ͨͳ͍߹ʹΘ͚ͯߟ͑Δ |G| |G| = 3 G = C |G| ≥ 4 C vw
ݭΛ࣋ͭ߹ ▸ ݭΛ࣋ͭͱ͖ด࿏ ͷΑ͏ʹ2ͭͷด࿏Λ࣋ͭ ▸ ͦΕͧΕͷด࿏ͷڥքͱ෦ͷͰͰ͖ΔάϥϑΛ ͱ͢Δ ▸ ͜ͷͱ͖
ͱͯ͠ҰൠੑΛࣦΘͳ͍ ▸ ԾఆΑΓ ͳͷͰ ͷ࠼৭͕Ͱ͖Δ ▸ طʹ৭͕ృΒΕ͍ͯΔͷͰԾఆΛద༻ͯ͠ ࠼৭Ͱ͖Δ ▸ ͕࠼৭Ͱ͖ͨ C1 , C2 ∈ C + vw v1 v2 ∈ C1 , v1 v2 ∉ C2 C G1 , G2 |G| > |G1 |, |G2 | G1 v, w G2 G
ݭΛ࣋ͭ߹
ݭΛ࣋ͨͳ͍߹ ▸ ͷྡΛɼ࣌ܭճΓʹ ͱஔ͘ ▸ ͷྖҬʹؚ·ΕɼྖҬࡾ֯ܗͳͷͰ ύε ▸ ͦΕΛ
ͱ͢Δͱ ด࿏ʹͳΔ vk v1 , u1 , ⋯, um , vk−1 ui C v1 , u1 , ⋯, um , vk−1 P C′ := P ∪ (C − vk )
ݭΛ࣋ͨͳ͍߹ ▸ ͷϦετ͔ΒҟͳΔ2৭ Λબͼɼ ͷϦετ͔Βআ ▸ ͷ֤ͷϦετʹ3৭Ҏ্͍ͬͯͯԾఆΑΓ ෦Λ࠼৭Ͱ͖Δ ▸ ʹ
ͷɼ ʹృΒΕͯͳ͍ํΛృΔ ▸ 1ྡ͍ͯ͠Δ ʹృΒΕ͍ͯΔͨΊ ʹͰ͖ͳ͔ͬͨ ▸ ʹృΒΕͨ߹ͲͪΒృΕͳ͘ͳΔՄೳੑ͕͋ΔͨΊআͨ͠ vk j, l ≠ 1 ui C′ C′ vk vk−1 j, l v1 j, l ui
͓ΘΓ ▸ ݭΛ࣋ͭͱ͖࣋ͨͳ͍ͱ͖ͷͲͪΒͰ5-Ϧετ࠼৭Մೳ ▸ Ϧετ࠼৭ͷ΄͏͕͍͠ͷͰવී௨ͷ5-࠼৭͕Մೳ ▸ ͬͨͶ ▸ 4-Ϧετ࠼৭ෆՄೳͰ͋Δ͜ͱ͕ΒΕ͍ͯΔΒ͍͠ ▸
ؤுͬͯྫ͔ূ໌Λߟ͑ͯΈ͍ͯͩ͘͞ɼ͍Ͱ͢ ▸ ݭΛ࣋ͨͳ͍߹͋ͨΓͰϠόͦ͏
͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂ