A basic elementary extension of the Duchet–Meyniel theorem

The Conjecture of Hadwiger implies that the Hadwiger number hh times the independence number αα of a graph is at least the number of vertices nn of the graph. In 1982 Duchet and Meyniel [P. Duchet, H. Meyniel, On Hadwiger’s number and the stability number, Ann. of Discrete Math. 13 (1982) 71–74] proved a weak version of the inequality, replacing the independence number αα by 2α−12α−1, that is, (2α−1)⋅h≥n.(2α−1)⋅h≥n. In 2005 Kawarabayashi, Plummer and the second author [K. Kawarabayashi, M. Plummer, B. Toft, Improvements of the theorem of Duchet and Meyniel on Hadwiger’s Conjecture, J. Combinatorial Theory B 95 (2005) 152–167] published an improvement of the theorem, replacing 2α−12α−1 by 2α−3/22α−3/2 when αα is at least 3. Since then a further improvement by Kawarabayashi and Song has been obtained, replacing 2α−12α−1 by 2α−22α−2 when αα is at least 3.In this paper a basic elementary extension of the Theorem of Duchet and Meyniel is presented. This may be of help to avoid dealing with basic cases when looking for more substantial improvements. The main unsolved problem (due to Seymour) is to improve, even just slightly, the theorem of Duchet and Meyniel in the case when the independence number αα is equal to 2. The case α=2α=2 of Hadwiger’s Conjecture was first pointed out by Mader as an interesting special case.