Three coloring of pure children drawings of snarks and the problem “The hunting of the snark”

T. E. Krenkel, T. A. Kulikova

Moscow Technical University of Communications and Informatics

Abstract: The Tait theorem as a consequence of Four Color Theorem states that planar qubic graphs are edge three-colorable. The first graph which is a counterexample for the Tait theorem was a nontrivial qubic (trivalent) Petersen graph $P,$ which is the only and minimal graph with chromatic index 4. The integer sequence OEIS A130315 describes (as defined by Martin Gardner) the number of (with girth $\ge5$) snarks, that are nontrivial qubic graphs with $2n$ verticies. The conjecture is presented that via transition from category Snarks to category SnarksPureDessins the derived two-colored graphs (pure children drawings of snarks) can be three-colored at halfedges. The embedding of the Petersen graph in double torus $\Sigma_2$ is presented. The $RGB$ theorem about the cycle double cover of the Peterson-Belyi graph $PB$ is proved.

UDC: 511.6+519.17

Received: 14.01.2016