Representation of proofs by colored graphs and the hadwiger conjecture

Abstract

A tinted graph is a graph whose arcs are colored with certain colors. A colored graph is a graph whose vertices are colored with certain colors. If M is the set of tinted (or colored tinted) graphs of order k and G is a tinted (or colored tinted) graph, then we shall say that G is M-regular (or M-regularly colored) if all its subgraphs of order k belong to M. We shall show how, for any formula p of the first-order predicate calculus, to construct a finite set B_p of tinted graphs of order 3 and a finite set C_p of colored tinted graphs of order 2 such that ¦-p if and only if there exists a B_p-regular tinted graph not admitting a C_p-regular coloring. Hadwiger's conjecture (HC) is as follows: If no subgraph of a graph without loops G is contractible to a complete graph of order n, then the vertices of G can be colored in n−1 colors such that neighboring vertices are colored with different colors. We construct a formula X of the first-order predicate calculus such that HC is equivalent with ⌉⊢X. Thus, HC reduces to HC₁: if all subgraphs of order 3 of the tinted graph G belong to B_X, then G is C_X-regularly colorable. Here B_X and C_X are specific finite sets of tinted graphs of order 3 and colored tinted graphs of order 2, respectively.