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Forcing a sparse minor

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  16 April 2015

BRUCE REED  and
DAVID R. WOOD
BRUCE REED
Affiliation:
School of Computer Science, McGill University, Montreal, H3A 0E9, Canada and National Institute of Informatics, Japan (e-mail: [email protected])
DAVID R. WOOD
Affiliation:
School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia (e-mail: [email protected])
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Abstract

This paper addresses the following question for a given graph H: What is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) ⩽ t + 6.291q (where the coefficient of 1 in the t term is best possible).

MSC classification

Primary: 05C83: Graph minors
Secondary: 05C35: Extremal problems 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 300 - 322
Copyright
Copyright © Cambridge University Press 2015 

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References

[1] Azuma, K. (1967) Weighted sums of certain dependent random variables. Tôhoku Math. J. (2) 19 357–367.CrossRefGoogle Scholar
[2] Bollobás, B., Catlin, P. A. and Erdős, P. (1980) Hadwiger's conjecture is true for almost every graph. European J. Combin. 1 195–199.Google Scholar
[3] Chudnovsky, M., Reed, B. and Seymour, P. (2011) The edge-density for K 2,t minors. J. Combin. Theory Ser. B 101 18–46.CrossRefGoogle Scholar
[4] Corradi, K. and Hajnal, A. (1963) On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hungar. 14 423–443.Google Scholar
[5] Diestel, R. (2010) Graph Theory , fourth edition, Vol. 173 of Graduate Texts in Mathematics, Springer.Google Scholar
[6] Dirac, G. A. (1964) Homomorphism theorems for graphs. Math. Ann. 153 69–80.CrossRefGoogle Scholar
[7] Fernandez de la Vega, W. (1983) On the maximum density of graphs which have no subcontraction to Ks . Discrete Math. 46 109–110.CrossRefGoogle Scholar
[8] Fox, J. and Sudakov, B. (2009) Density theorems for bipartite graphs and related Ramsey-type results. Combinatorica 29 153–196.CrossRefGoogle Scholar
[9] Jørgensen, L. K. (1994) Contractions to K 8. J. Graph Theory 18 431–448.CrossRefGoogle Scholar
[10] Justesen, P. (1989) On independent circuits in finite graphs and a conjecture of Erdős and Pósa. Ann. Discrete Math. 41 299–305.CrossRefGoogle Scholar
[11] Kostochka, A. V. (1982) The minimum Hadwiger number for graphs with a given mean degree of vertices. Metody Diskret. Analiz. 38 37–58.Google Scholar
[12] Kostochka, A. V. (1984) Lower bound of the Hadwiger number of graphs by their average degree. Combinatorica 4 307–316.Google Scholar
[13] Kostochka, A. V. and Prince, N. (2008) On Ks,t -minors in graphs with given average degree. Discrete Math. 308 4435–4445.Google Scholar
[14] Kostochka, A. V. and Prince, N. (2010) Dense graphs have K 3,t minors. Discrete Math. 310 2637–2654.CrossRefGoogle Scholar
[15] Kostochka, A. V. and Prince, N. (2012) On Ks,t -minors in graphs with given average degree, II. Discrete Math. 312 3517–3522.Google Scholar
[16] Kühn, D. and Osthus, D. (2005) Forcing unbalanced complete bipartite minors. European J. Combin. 26 75–81.CrossRefGoogle Scholar
[17] Mader, W. (1967) Homomorphieeigenschaften und mittlere Kantendichte von Graphen. Math. Ann. 174 265–268.CrossRefGoogle Scholar
[18] Mader, W. (1968) Homomorphiesätze für Graphen. Math. Ann. 178 154–168.CrossRefGoogle Scholar
[19] Mader, W. (1972) Existenz n-fach zusammenhängender Teilgraphen in Graphen genügend grosser Kantendichte. Abh. Math. Sem. Univ. Hamburg 37 86–97.Google Scholar
[20] Myers, J. S. (2002) Graphs without large complete minors are quasi-random. Combin. Probab. Comput. 11 571–585.CrossRefGoogle Scholar
[21] Myers, J. S. (2003) The extremal function for unbalanced bipartite minors. Discrete Math. 271 209–222.Google Scholar
[22] Myers, J. S. and Thomason, A. (2005) The extremal function for noncomplete minors. Combinatorica 25 725–753.CrossRefGoogle Scholar
[23] Song, Z. (2004) Extremal Functions for Contractions of Graphs. PhD thesis, Georgia Institute of Technology, USA.Google Scholar
[24] Song, Z. X. and Thomas, R. (2006) The extremal function for K 9 minors. J. Combin. Theory Ser. B 96 240–252.Google Scholar
[25] Thomas, R. and Wollan, P. (2005) An improved linear edge bound for graph linkages. European J. Combin. 26 309–324.Google Scholar
[26] Thomason, A. (1984) An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc. 95 261–265.CrossRefGoogle Scholar
[27] Thomason, A. (2001) The extremal function for complete minors. J. Combin. Theory Ser. B 81 318–338.CrossRefGoogle Scholar
[28] Thomason, A. (2007) Disjoint complete minors and bipartite minors. European J. Combin. 28 1779–1783.Google Scholar
[29] Thomason, A. (2008) Disjoint unions of complete minors. Discrete Math. 308 4370–4377.Google Scholar
[30] Verstraëte, J. (2002) A note on vertex-disjoint cycles. Combin. Probab. Comput. 11 97–102.Google Scholar