Combinatorial operations on near-triangulations of the plane.

*(Chinese. English summary)*Zbl 0789.05038Summary: In this paper combinatorial operations, \(T^*\), \(T^ +\) and \(\pi\), on near-triangulations are introduced and used in a process of building up a given near-triangulation \(G\) bounded by a circuit \(Q_ r\). In this process one starts from an arbitrary triangle \(\Delta\), and adds a new triangle \(\Delta_{i+1}\), at each time, to the intermediate near- triangulation \(G_ i\) previously formed so that one or two properly assigned sides on the bounding circuit of \(G_ i\) is or are coincident with that of \(\Delta_{i+1}\). At the end of this process one gets \(G\).

Based on the above combinatorial results, conjectures which are concerned only with the properties of 4-colorings of circuits and each of which is equivalent to the Four-Color Theorem are given in the present paper. It is also pointed out that an enlightening conjecture of the above type—a conjecture at the end of the paper [H. Whitney and W. T. Tutte, Util. Math. 2, 241-281 (1972; Zbl 0253.05120)] is not true even for circuits of length 4.

Based on the above combinatorial results, conjectures which are concerned only with the properties of 4-colorings of circuits and each of which is equivalent to the Four-Color Theorem are given in the present paper. It is also pointed out that an enlightening conjecture of the above type—a conjecture at the end of the paper [H. Whitney and W. T. Tutte, Util. Math. 2, 241-281 (1972; Zbl 0253.05120)] is not true even for circuits of length 4.