A reformulation of Hadwiger’s conjecture

Let G = (V,E) be a finite, simple, undirected graph. Hadwiger’s conjecture states that

if n = \chi(G) then K_n is a minor of G.

Let’s call that statement (Hadw). We show that it is equivalent to the following statement:

(Hadw’): If G is not a complete graph, then there is a minor M of G such that

  1. M \not \cong G, and
  2. \chi(M) = \chi(G).

It is clear that (Hadw) implies (Hadw’). For the other direction, take any finite graph G and let n = \chi(G). If G is complete, we are done. If not, use (Hadw’) and let M_1 be a proper minor such that \chi(M_1) = n. If M_1 is complete, we are done, otherwise use (Hadw’) again to get a proper minor M_2 of M_1 with \chi(M_2) =n. And so on. Since G is finite, this procedure is bound to end at M_k for some k\in\mathbb{N}. We
have \chi(M_k) = n, and since the procedure ended at k, the graph M_k must be complete. So we get statement (Hadw).


About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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