and Thomas)

Robertson, Seymour and Thomas in a paper of 146 pages long (see [1]); in this manuscript, via an original speech and simple results, we rigorously simplify the understanding of this solved 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. We say that a graph B is berge (see [4]) if every graph B ′ ∈ {B, B̄} does not contain an induced cycle of odd length ≥ 5 (B ̄ is the complementary graph of B). A graph G is perfect if every induced subgraph G ′ of G satisfies χ(G′) = ω(G′), where χ(G′) is the chromatic number of G ′ and ω(G′) is the clique number of G′. The Berge conjecture states that a graph H is perfect if and only if H is berge. Indeed, the difficult part of the Berge conjecture consists to show that χ(B) = ω(B) for every berge graph B (Briefly, the difficult part of the Berge conjecture will be called the Berge problem (see [4])). The Hadwiger conjecture (see [4]), states that every graph G satisfies χ(G) ≤ η(G) (where η(G) is th