A new generalization of subgraphs whose edges have distinct colors and its applications to BH conjecture

• Project/Area Number: 24740068
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kochi University
• Principal Investigator: Suzuki kazuhiro 高知大学, 教育研究部自然科学系, 助教 (50514410)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: グラフ理論 / 離散幾何学 / 全域木 / 辺着色 / 辺彩色 / 異色全域木 / 異色部分グラフ / BH予想 / (g,f)-異色部分グラフ / f-異色全域木 / f-辺彩色 / 次数和 / 分割 / 単色全域木

Outline of Final Research Achievements

A subgraph whose edges have distinct colors is called a heterochromatic subgraph. We define a (g,f)-chromatic subgraph as a subgraph having at least g(c) and at most f(c) edges colored with c for any color c, where g and f are functions from a color set to the set of non-negative integers. In this research, we studied conditions for existence of a (g,f)-chromatic subgraph in edge-colored graphs, and we got the following results.
(1) We got a necessary and sufficient condition for existence of a (g,f)-chromatic subgraph in edge-colored graphs. (2) We studied heterochromatic subgraphs with some upper bound of maximum degree. (3) We generalized some previous results on heterochromatic subgraphs. (4) We got a sufficient condition for existence of a spanning k-tree in bipartite graphs. (5) We got a Lemma for balanced partition of a sequence of colored elements.

Research Products

• [Journal Article] Spanning k-trees of Bipartite Graphs (2015)
  • Author(s): Kano, Mikio; Ozeki, Kenta; Suzuki, Kazuhiro; Tsugaki, Masao
  • Journal Title: The Electronic Journal of Combinatorics Volume: 22
  • Peer Reviewed / Open Access
• [Journal Article] An f-chromatic spanning forest of edge-colored complete bipartite graphs (2015)
  • Author(s): Suzuki, Kazuhiro
  • Journal Title: The Australasian Journal of Combinatorics Volume: 61 Pages: 130-137
  • Peer Reviewed / Open Access
• [Journal Article] Properly colored geometric matchings and 3-trees without crossings on multicolored points in the plane (2014)
  • Author(s): Kano, Mikio; Suzuki, Kazuhiro; Uno, Miyuki
  • Journal Title: Discrete and Computational Geometry and Graphs: 16th Japanese Conference, JCDCGG 2013, Tokyo, Japan, September 17-19, 2013, Revised Selected Papers, LNCS Volume: 8845 Pages: 96-111
  • DOI: 10.1007/978-3-319-13287-7_9
  • ISBN: 9783319132860, 9783319132877
  • Peer Reviewed
• [Journal Article] A Generalization of Heterochromatic Graphs and f-Chromatic Spanning Forests. Graphs and Combinatorics (2013)
  • Author(s): Suzuki, Kazuhiro
  • Journal Title: Graphs and Combinatorics Volume: Vol.29, No.3 Issue: 3 Pages: 715-727
  • DOI: 10.1007/s00373-011-1125-z
  • Peer Reviewed
• [Presentation] Properly colored geometric matchings and 3-trees without crossings on multicolored points in the plane. (2013)
  • Author(s): Kano, Mikio; Suzuki, Kazuhiro; Uno, Miyuki
  • Organizer: 16th Japan Conference on Discrete and Computational Geometry and Graphs (JCDCG2 2013)
  • Place of Presentation: 東京理科大学
• [Presentation] (g,f)-chromatic forests (2012)
  • Author(s): 鈴木一弘
  • Organizer: 離散数学とその応用研究集会2012
  • Place of Presentation: 茨城大学日立キャンパス