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五色定理の証明

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September 03, 2018

 五色定理の証明

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pazzle1230

September 03, 2018
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  1. ༻ޠ ▸ Ϧετ࠼৭ ▸ ɹɹɹ ͕༩͑ΒΕͨͱ͖͢΂ͯͷ ʹରͯ͠ ▸ બ୒਺(Ϧετ࠼৭਺): ͱද͢

    ▸ Ͱ͋Δάϥϑ͢΂ͯͰϦετ࠼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)
  2. Ϧετ࠼৭Λ࢖Θͳ͍ূ໌ ▸ ͬͪ͜͸ϝΠϯ͡Όͳ͍ͨΊ͓ؾ࣋ͪײΛߴ͍ͯ͘͘͠ ▸ ฏ໘ࡾ֯ܗ෼ׂ͠EulerͷެࣜͰؤுΔͱ ͕Ͷ ▸ ࣍਺5ҎԼͷ௖఺ ͕ଘࡏ͢Δ ▸

    nͷؼೲ๏Ͱূ໌͢ΔͷͰ ͸ࣗ໌ʹ੒ཱɼ Ͱ΍Δ m ≤ 3n − 6 d(G) = 2m/n ≤ 2(3n − 6)/n < 6 n ≤ 5 n > 5 v
  3. ͓ؾ࣋ͪূ໌(ূ໌Ͱ͸ͳͦ͞͏) Ϧετ࠼৭Λ࢖Θͳ͍ূ໌ ▸ 5ͭͷྡ઀௖఺Λ࣌ܭճΓʹ ͱ͠ɼ৭΋ ▸ ͕1ͱ3ͷ௖఺ͷΈ͔ΒͳΔύεͰܨ͕ΕͯΔͱԾఆ ▸ ඇ࿈݁ͳΒ Λ

    ͱಉ৭ʹؤுͬͯͰ͖͖ͯͬ͞ͱҰॹ ▸ ͷ࿈݁੒෼Ͱ1ͱ3Λަ׵ͨ͠Β ͷ৭͸มΘΒͳ͍ͨΊ v1 , ⋯, v5 v1 , v3 v3 v1 v1 1,⋯,5 v3
  4. Ϧετ࠼৭ʹΑΔূ໌ ▸ ࣌ؒతʹ΍ͬͺΓ͓ؾ࣋ͪ ▸ ࣍ͷओுΛূ໌͢Δ ▸ خ͍͠ཧ༝ ▸ ͷۃେฏ໘άϥϑΛߟ͑Δͱ֎ྖҬΛؚΉશ͕ͯࡾ֯ܗͰ෼ׂ ▸

    ֎ྖҬͱͷڥքʹؚ·ΕΔ3௖఺ͷ͏ͪɼ2ͭΛృΓ Λద༻Ͱ͖Δ શͯͷ಺ྖҬ͕ࡾ֯ܗͰɼ֎ྖҬͱͷڥքʹؚ·ΕΔด࿏ ͕͋ΔͱԾఆ͢Δɽ௖఺ ʹ͸ͦΕͧΕ1,2ΛృΓɼ ͷ֤௖఺ʹ͸3৭Ҏ্ͷ Ϧετɼ ͷ֤௖఺ʹ͸5৭Ҏ্ͷϦετ͕༩͑ΒΕ͍ͯΔͱ͖ɼ ͷ࠼৭ Λ֦ு͠ ͷ࠼৭͕ߏ੒Ͱ͖Δ C = v1 ⋯vk v1 v1 , v2 C G − C v1 , v2 G G ( * ) ( * )
  5. ݭΛ࣋ͭ৔߹ ▸ ݭΛ࣋ͭͱ͖ด࿏ ͸ ͷΑ͏ʹ2ͭͷด࿏Λ࣋ͭ ▸ ͦΕͧΕͷด࿏ͷڥքͱ಺෦ͷ௖఺ͰͰ͖ΔάϥϑΛ 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
  6. ݭΛ࣋ͨͳ͍৔߹ ▸ ͷྡ઀఺Λɼ࣌ܭճΓʹ ͱஔ͘ ▸ ͸ ͷ಺ྖҬʹؚ·Εɼ಺ྖҬ͸ࡾ֯ܗͳͷͰ ͸ύε ▸ ͦΕΛ

    ͱ͢Δͱ ͸ด࿏ʹͳΔ vk v1 , u1 , ⋯, um , vk−1 ui C v1 , u1 , ⋯, um , vk−1 P C′ := P ∪ (C − vk )
  7. ݭΛ࣋ͨͳ͍৔߹ ▸ ͷϦετ͔ΒҟͳΔ2৭ Λબͼɼ ͷϦετ͔Β࡟আ ▸ ͷ֤௖఺ͷϦετʹ͸3৭Ҏ্࢒͍ͬͯͯԾఆΑΓ ಺෦Λ࠼৭Ͱ͖Δ ▸ ʹ͸

    ͷ಺ɼ ʹృΒΕͯͳ͍ํΛృΔ ▸ 1͸ྡ઀͍ͯ͠Δ ʹృΒΕ͍ͯΔͨΊ ʹ͸Ͱ͖ͳ͔ͬͨ ▸ ʹృΒΕͨ৔߹͸ͲͪΒ΋ృΕͳ͘ͳΔՄೳੑ͕͋ΔͨΊ࡟আͨ͠ vk j, l ≠ 1 ui C′ C′ vk vk−1 j, l v1 j, l ui