We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic.
Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K k -minor is ck-colorable.
Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Böhme et al.
We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) − 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.
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K. Appel and W. Haken: Every planar map is four colorable, Part I. Discharging; Illinois J. Math. 21 (1977), 429–490.
K. Appel, W. Haken and J. Koch: Every planar map is four colorable, Part II. Reducibility; Illinois J. Math. 21 (1977), 491–567.
T. Böhme, K. Kawarabayashi, J. Maharry and B. Mohar: Linear connectivity forces large complete bipartite minors, to appear in J. Combin. Theory Ser. B.
P. A. Catlin: A bound on the chromatic number of a graph, Discrete Math 22 (1978), 81–83.
R. Diestel: Graph Theory, Graduate texts in mathematics 173, Springer (1997).
G. A. Dirac: A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92.
P. Duchet and H. Meyniel: On Hadwiger’s number and the stability number, Annals of Discrete Math. 13 (1982), 71–74.
H. Hadwiger: Über eine Klassifikation der Streckenkomplexe, Vierteljahrsschr. Naturforsch, Ges. Zürich 88 (1943), 133–142.
J. Geelen, B. Gerards, L. Goddyn, B. Reed, P. Seymour and A. Vetta: The odd case of Hadwiger’s conjecture, submitted.
B. Guenin: Talk at Oberwolfach on Graph Theory, Jan. 2005.
T. R. Jensen and B. Toft: Graph Coloring Problems, Wiley-Interscience, 1995.
K. Kawarabayashi: Minors in 7-chromatic graphs, preprint.
K. Kawarabayashi and B. Toft: Any 7-chromatic graph has K 7 or K 4,4 as a minor, Combinatorica 25(3) (2005), 327–353.
K. Kawarabayashi, M. Plummer and B. Toft: Improvements of the theorem of Duchet and Meyniel on Hadwiger’s Conjecture, J. Combin. Theory Ser. B 95 (2005), 152–167.
K. Kawarabayashi and Z. Song: Independence number and clique minors, submitted.
A. Kostochka: The minimum Hadwiger number for graphs with a given mean degree of vertices (in Russian), Metody Diskret. Analiz. 38 (1982), 37–58.
A. Kostochka: Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica 4 (1984), 307–316.
M. D. Plummer, M. Stiebitz and B. Toft: On a special case of Hadwiger’s Conjecture, Discuss. Math. Graph Theory 23 (2003), 333–363.
B. Reed and P. D. Seymour: Fractional colouring and Hadwiger’s conjecture, J. Combin. Theory Ser. B 74 (1998), 147–152.
N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas: The four-color theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.
N. Robertson, P. D. Seymour and R. Thomas: Hadwiger’s conjecture for K 6-free graphs, Combinatorica 13 (1993), 279–361.
A. Thomason: An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984), 261–265.
A. Thomason: The extremal function for complete minors, J. Combin. Theory Ser. B 81 (2001), 318–338.
K. Wagner: Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), 570–590.
D. R. Woodall: Subcontraction-equivalence and Hadwiger’s Conjecture, J. Graph Theory 11 (1987), 197–204.
Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by Sumitomo Foundation and by Inoue Research Award for Young Scientists.
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Kawarabayashi, Ki., Song, ZX. Some remarks on the odd hadwiger’s conjecture. Combinatorica 27, 429 (2007). https://doi.org/10.1007/s00493-007-2213-9
Mathematics Subject Classification (2000)