グラフ彩色問題における4色定理の応用、拡張に関する研究

• Project/Area Number: 16740044
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: National Institute of Informatics (2006); Tohoku University (2004-2005)
• Principal Investigator: 河原林 健一 国立情報学研究所, 情報学プリンシプル研究系, 助教授 (40361159)
• Project Period (FY): 2004 – 2006
• Project Status: Completed (Fiscal Year 2006)
• Budget Amount: ¥3,700,000 (Direct Cost: ¥3,700,000); Fiscal Year 2006: ¥1,200,000 (Direct Cost: ¥1,200,000); Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000); Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
• Keywords: 4色定理 / Hadweiger予想 / グラフマイナー / 曲面上のグラフ / グラフ彩色 / リスト彩色 / グラフ理論 / アルゴリズム / HADWIGER予想 / 理想グラフ / グラフ構造理論 / グラフ連結度 / グラフマイナー理論 / Hadwiger予想

Research Abstract

この3年間で次のような研究を行った。
1.4色定理の曲面上への拡張に関する研究を行い、MOHARと5リスト彩色定理を証明した。これはThomassenの5色定理の拡張である。
2.Thomassenと平面グラフに関する重要な結果、Grotszchの定理の拡張に成功した。また曲面上への応用も導いた。
3.平面グラフ、曲面上のグラフの拡張である「マイナー」に関して閉じたグラフの族に対して、グラフ彩色問題のアルゴリズム的解析を行い、従来知られていた結果の大幅な改良に成功し、この結果は、理論計算機最高の会議に論文が採録された。
4.Hadwiger予想に関する重要な未解決問題、Jorgensen予想をR.Thomasらとの共同研究で、巨大なグラフにおいては、完全解決をした。
5.Hadwiger予想の拡大予想であるODD Hadwiger予想に対して、漸近的な解決を与えた。現在まで、上界すら知られていたなかった中で大きな進歩である。また、証明のアルゴリズム的応用も行った。
6.グラフ交差数等いう平面グラフから派生したグラフの重要な普遍量に対して、アルゴリズム面では、平面グラフの拡張に成功した。
7.Robertson-SeymourのThe disjoint paths problemの拡張を、B.Reedとともに行い、パリティバージョンへの拡張に成功した。
以上の7つの研究を中心に行った。

Research Products

• [Journal Article] On the connectivity of mininmum and minimal counterexamples to Hadwiger's Conjecture (2007)
  • Author(s): K.Kawarabayashi
  • Journal Title: J. Combin. Theory Ser. B 97 Pages: 144-150
• [Journal Article] $2$-connected spanning subgraphs with low maximum degree in locally planar graphs (2007)
  • Author(s): M.Ellingham, K.Kawarabayashi
  • Journal Title: J. Combin. Theory Ser. B 97 Pages: 401-412
• [Journal Article] A relaxed version of Hadwiger's conjecture for list-coloring (2007)
  • Author(s): K.Kawarabayashi, B.Mohar
  • Journal Title: J. Combin. Theory Ser. B 97 Pages: 647-651
• [Journal Article] Some recent progress and applications on Graph Minor Theory (2007)
  • Author(s): K.Kawarabayashi, B.Mohar
  • Journal Title: Graphs and Combinatorics 23 Pages: 1-46
• [Journal Article] Computing crossing number in linear time (2007)
  • Author(s): K.Kawarabayashi, B.Reed
  • Journal Title: ACM Symposium on Theory of Computing (STOC' 07) 39 Pages: 382-390
• [Journal Article] The Erdos-Posa property for orientable surface (2007)
  • Author(s): K.Kawarabayashi, A.Nakamoto
  • Journal Title: Discrete Math. 307 Pages: 764-768
• [Journal Article] Half integral packing, Erd＆odblac;s-Posa propert and Graph minors (2007)
  • Author(s): K.Kawarabayashi
  • Journal Title: ACM-SIAM Symposium on Discrete Algorithms 2007 (SODA) 18 Pages: 1187-1196
• [Journal Article] Rooted minor problems in graphs (2007)
  • Author(s): L.Jorgensen, K.Kawarabayashi
  • Journal Title: J. Graph Theory 55 Pages: 191-207
• [Journal Article] Chords of longest circuits in locally planar graphs (2007)
  • Author(s): K.Kawarabayashi, J.Niu, C.Q.Zhang
  • Journal Title: Europ. J. Combinatorics 28 Pages: 315-321
• [Journal Article] Dominating number in cubic graphs with large girth (2006)
  • Author(s): K.Kawarabayashi, M.Plummer, A.Saito
  • Journal Title: J. Graph Theory 52 Pages: 1-6
• [Journal Article] Non-zero disjoint cycles in highly connected graphs (2006)
  • Author(s): K.Kawarabayashi, P.Wollan
  • Journal Title: J. Combin. Theory Ser. B 96 Pages: 296-301
• [Journal Article] Approximating the chromatic number and the list-chromatic number of minor-closed family (2006)
  • Author(s): K.Kawarabayashi, B.Mohar
  • Journal Title: ACM Symposium on Theory of Computing (STOC' 06) 38 Pages: 401-416
• [Journal Article] On sufficient degree conditions for a graph to be $k$-linked (2006)
  • Author(s): K.Kawarabayashi, A.Kostochka, G.Yu
  • Journal Title: Lombinatorics, Probability and Computing 15 Pages: 685-894
• [Journal Article] Detecting Even Holes (2005)
  • Author(s): M.Chudnovsky
  • Journal Title: J.Graph Theory 48 Pages: 85-111
• [Journal Article] Any $7$-chromatic graph has a $K_7$-minor or a $K_{4,4}$-minor (2005)
  • Author(s): K.Kawarabayashi, B.Toft
  • Journal Title: Combinatorica 25 Pages: 327-353
• [Journal Article] Improvement of the theorem of Duchet and Meyniel on Hadwiger's Conjecture (2005)
  • Author(s): K.Kawarabayashi
  • Journal Title: J.Combin.Theory Ser.B 95 Pages: 152-167
• [Journal Article] Algorithmic graph minor theory: Decomposition, Approximation and Coloring (2005)
  • Author(s): E.Demaine
  • Journal Title: Proceedings of the 46th Annual Symposium on Foundations of Computer Science (FOCS) 46 Pages: 637-646
• [Journal Article] Nonseparating paths with two prescribed endvertices in $4$-connected graphs (2005)
  • Author(s): K.Kawarabayashi
  • Journal Title: Annals of Combinatorics 9 Pages: 47-56
• [Journal Article] On Structure of $k$-connected graphs without $K_k$-minors (2005)
  • Author(s): K.Kawarabayashi
  • Journal Title: Europ.J.Combinatorics 26 Pages: 293-310
• [Journal Article] Graph Minors and Linkages (2005)
  • Author(s): G.Chen
  • Journal Title: J.Graph Theory 49 Pages: 75-91
• [Journal Article] Some property of 5-contraction-critical graphs (2005)
  • Author(s): K.Ando
  • Journal Title: Graphs and Combinatorics 21 Pages: 27-37
• [Journal Article] Acute triangles in 4-connected plane graphs (2005)
  • Author(s): K.Kawarabayashi
  • Journal Title: Discrete Math. 292 Pages: 95-106
• [Journal Article] Existence of two long cycles (2005)
  • Author(s): Y.Egawa
  • Journal Title: Discrete Math 305 Pages: 154-169
• [Journal Article] Detecting Even Holes (2005)
  • Author(s): M.Chudnovsky, K.Kawarabayashi, P.Seymour
  • Journal Title: Journal of Graph Theory 48(2) Pages: 85-111
• [Journal Article] K-Linked Graphs with Girth Condition (2004)
  • Author(s): K.Kawarabayashi
  • Journal Title: J.Graph Theory 45(1) Pages: 48-50
• [Journal Article] Cycles through prescribed vertex set in n-connected graphs (2004)
  • Author(s): K.Kawarabayashi
  • Journal Title: J.Combin.Theroy Ser.B. 90(2) Pages: 315-323
• [Journal Article] A theorem on paths in locally triangulations. (2004)
  • Author(s): K.Kawarabayashi
  • Journal Title: Europ.J.Combinatorics 25 Pages: 781-784
• [Journal Article] Rooted minors problem in highly connected graphs (2004)
  • Author(s): K.Kawarabayashi
  • Journal Title: Discrete Math. 287 Pages: 121-123
• [Journal Article] Vertex-Disjoint Cycles Containing Specified Vertices in a Bipartite Graph (2004)
  • Author(s): G.Chen, H.Enomoto, K.Kawarabayashi, D.Lou, K.Ota, A.Saito
  • Journal Title: Journal of Graph Theory 46(3) Pages: 145-166