Connected matchings and Hadwiger's conjecture

Conjecture 0.3 has a fairly interesting reformulation with some ”Ramsey flavor”. A set of pairwise disjoint edges e1, e2, . . . , et of G is called a connected matching of size t ([8]) if for every 1 ≤ i < j ≤ t there exists at least one edge of G connecting an endpoint of ei to an endpoint of ej . Alfréd Rényi Institute, Hungarian Academy of Sciences, Budapest 1364, P. O. Box 127 and Dept. of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA. Email: [email protected] Research partially supported by Hungarian National Science Foundation Grants OTKA T032452 and T037846 and by NSF grant DMS 0140692 Computer and Automation Research Institute of the Hungarian Academy of Sciences, Budapest, P. O. Box 63, Hungary-1518. Email: [email protected] Alfréd Rényi Institute, Hungarian Academy of Sciences, Budapest 1364, P. O. Box 127. Email: [email protected] Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846 and T046376.