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2011 | 31 | 4 | 639-674
Tytuł artykułu

Complete minors, independent sets, and chordal graphs

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) ≥ χ(G). Since χ(G) α(G) ≥ |V(G)|, Hadwiger's Conjecture implies that α(G) h(G) ≥ |V(G)|. We show that (2α(G) - ⌈log_{τ}(τα(G)/2)⌉) h(G) ≥ |V(G)| where τ ≍ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2α(G) - 2) h(G) ≥ |V(G) | when α(G) ≥ 3.
Wydawca
Rocznik
Tom
31
Numer
4
Strony
639-674
Opis fizyczny
Daty
wydano
2011
otrzymano
2009-09-14
poprawiono
2010-09-15
zaakceptowano
2010-10-06
Twórcy
  • Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
  • Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA
autor
  • Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
autor
  • Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
Bibliografia
  • [1] K. Appel and W. Haken, Every planar map is four colorable. I. Discharging, Illinois J. Math. 21 (1977) 429-490.
  • [2] K. Appel, W. Haken and J. Koch, Every planar map is four colorable. II. Reducibility, Illinois J. Math. 21 (1977) 491-567.
  • [3] J. Balogh, J. Lenz and H. Wu, Complete Minors, Independent Sets, and Chordal Graphs, http://arxiv.org/abs/0907.2421.
  • [4] C. Berge, Les problemes de coloration en theorie des graphes, Publ. Inst. Statist. Univ. Paris 9 (1960) 123-160.
  • [5] M. Chudnovsky and A. Fradkin, An approximate version of Hadwiger's conjecture for claw-free graphs, J. Graph Theory 63 (2010) 259-278.
  • [6] G. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952) 85-92, doi: 10.1112/jlms/s1-27.1.85.
  • [7] G. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76, doi: 10.1007/BF02992776.
  • [8] P. Duchet and H. Meyniel, On Hadwiger's number and the stability number, in: Graph Theory (Cambridge, 1981), (North-Holland, Amsterdam, 1982) 71-73.
  • [9] J. Fox, Complete minors and independence number, to appear in SIAM J. Discrete Math.
  • [10] H. Hadwiger, Uber eine Klassifikation der Streckenkomplexe, Vierteljschr. Naturforsch. Ges. Zürich 88 (1943) 133-142.
  • [11] K. Kawarabayashi, M. Plummer and B. Toft, Improvements of the theorem of Duchet and Meyniel on Hadwiger's conjecture, J. Combin. Theory (B) 95 (2005) 152-167, doi: 10.1016/j.jctb.2005.04.001.
  • [12] K. Kawarabayashi and Z. Song, Independence number and clique minors, J. Graph Theory 56 (2007) 219-226, doi: 10.1002/jgt.20268.
  • [13] L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267, doi: 10.1016/0012-365X(72)90006-4.
  • [14] F. Maffray and H. Meyniel, On a relationship between Hadwiger and stability numbers, Discrete Math. 64 (1987) 39-42, doi: 10.1016/0012-365X(87)90238-X.
  • [15] M. Plummer, M. Stiebitz and B. Toft, On a special case of Hadwiger's conjecture, Discuss. Math. Graph Theory 23 (2003) 333-363, doi: 10.7151/dmgt.1206.
  • [16] N. Robertson, D. Sanders, P. Seymour and R. Thomas, The four-colour theorem, J. Combin. Theory (B) 70 (1997) 2-44, doi: 10.1006/jctb.1997.1750.
  • [17] N. Robertson, P. Seymour and R. Thomas, Hadwiger's conjecture for K₆-free graphs, Combinatorica 13 (1993) 279-361, doi: 10.1007/BF01202354.
  • [18] B. Toft, A survey of Hadwiger's conjecture, Congr. Numer. 115 (1996) 249-283, Surveys in Graph Theory (San Francisco, CA, 1995).
  • [19] K. Wagner, Uber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937) 570-590, doi: 10.1007/BF01594196.
  • [20] D. Wood, Independent sets in graphs with an excluded clique minor, Discrete Math. Theor. Comput. Sci. 9 (2007) 171-175.
  • [21] D.R. Woodall, Subcontraction-equivalence and Hadwiger's conjecture, J. Graph Theory 11 (1987) 197-204, doi: 10.1002/jgt.3190110210.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1571
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