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Combinatorial operations on near-triangulations of the plane. (Chinese. English summary) Zbl 0789.05038
Summary: 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.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C99 Graph theory