Complete minors, independent sets, and chordal graphs

József Balogh ; John Lenz ; Hehui Wu

Discussiones Mathematicae Graph Theory, Tome 31 (2011), p. 639-674 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2011-01-01

Zbl 1255.05184

EUDML-ID : urn:eudml:doc:270983

Bibliographie

[000] [1] K. Appel and W. Haken, Every planar map is four colorable. I. Discharging, Illinois J. Math. 21 (1977) 429-490. | Zbl 0387.05009

[001] [2] K. Appel, W. Haken and J. Koch, Every planar map is four colorable. II. Reducibility, Illinois J. Math. 21 (1977) 491-567. | Zbl 0387.05010

[002] [3] J. Balogh, J. Lenz and H. Wu, Complete Minors, Independent Sets, and Chordal Graphs, http://arxiv.org/abs/0907.2421. | Zbl 1255.05184

[003] [4] C. Berge, Les problemes de coloration en theorie des graphes, Publ. Inst. Statist. Univ. Paris 9 (1960) 123-160. | Zbl 0103.16201

[004] [5] M. Chudnovsky and A. Fradkin, An approximate version of Hadwiger's conjecture for claw-free graphs, J. Graph Theory 63 (2010) 259-278. | Zbl 1216.05061

[005] [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. | Zbl 0046.41001

[006] [7] G. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76, doi: 10.1007/BF02992776. | Zbl 0098.14703

[007] [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. | Zbl 0522.05060

[008] [9] J. Fox, Complete minors and independence number, to appear in SIAM J. Discrete Math. | Zbl 1221.05289

[009] [10] H. Hadwiger, Uber eine Klassifikation der Streckenkomplexe, Vierteljschr. Naturforsch. Ges. Zürich 88 (1943) 133-142. | Zbl 0061.41308

[010] [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. | Zbl 1080.05074

[011] [12] K. Kawarabayashi and Z. Song, Independence number and clique minors, J. Graph Theory 56 (2007) 219-226, doi: 10.1002/jgt.20268. | Zbl 1135.05068

[012] [13] L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267, doi: 10.1016/0012-365X(72)90006-4. | Zbl 0239.05111

[013] [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. | Zbl 0628.05052

[014] [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. | Zbl 1053.05052

[015] [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. | Zbl 0883.05056

[016] [17] N. Robertson, P. Seymour and R. Thomas, Hadwiger's conjecture for K₆-free graphs, Combinatorica 13 (1993) 279-361, doi: 10.1007/BF01202354. | Zbl 0830.05028

[017] [18] B. Toft, A survey of Hadwiger's conjecture, Congr. Numer. 115 (1996) 249-283, Surveys in Graph Theory (San Francisco, CA, 1995). | Zbl 0895.05025

[018] [19] K. Wagner, Uber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937) 570-590, doi: 10.1007/BF01594196. | Zbl 63.0550.01

[019] [20] D. Wood, Independent sets in graphs with an excluded clique minor, Discrete Math. Theor. Comput. Sci. 9 (2007) 171-175. | Zbl 1153.05049

[020] [21] D.R. Woodall, Subcontraction-equivalence and Hadwiger's conjecture, J. Graph Theory 11 (1987) 197-204, doi: 10.1002/jgt.3190110210. | Zbl 0672.05027