A simple dissertation around the Berge problem and the Hadwiger conjecture

Abstract

The Hadwiger conjecture (recall (see [8]) that the famous four-color problem is a special case of the Hadwiger conjecture) states that every graph G satisfies χ(G) ≤ η(G) (where χ(G) is the chromatic number of G, and η(G) is the hadwiger number of G (i.e. the maximum of p such that G is contractible to the complete graph Kp )). The Berge problem was proved by Chudnovsky, Robertson, Seymour and Thomas in a paper of 146 pages long (see [1]). In this manuscript, via an original speech and results, we rigorously simplify the understanding of the Berge problem and the Hadwiger conjecture. It will appear that what Chudnovsky, Robertson, Seymour and Thomas were proved in their paper of 146 pages long, was an analytic conjecture stated in a very small class of graphs; it will also appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs. Moreover, we will see that the Hadwiger conjecture and the Berge problem are rigorously resembling. Based on this resemblance, it will become natural and not surprising to us to conjecture that the Hadwiger conjecture and the Berge conjecture are strongly related.