极大平面图的组合运算

COMBINATORIAL OPERATIONS ON NEAR-TRIANGULATIONS OF THE PLANE

一、引言有关平面4色问题的综合论述见文[1]和文[2].目前这一问题因有计算机辅助证明而得到解决,但这种证明繁杂.又由于等价的命题多,联系的方面广,因而对此问题进行新的理论探讨,便不会是毫无意义.把平面4色问题化成只与圈上的4染色集性质有关的问题来解决,这种思想在文[3]中已经有了,只可惜该文最后所提出的猜想不真(对长度为4的圈已不成立.此事我们将另文评述).本文以极大平面图的组合序列为基础,得出一些只与圈上的4染色集的性质有关的等价于平面4色猜想的命题.

In this paper combinatorial operations,T~*,T^+ and π,on near-triangulations are intro-duced and used in a process of building up a given near-triangulation G bounded by a circuitQ_r.In this process one starts from an arbitrary triangle Δ,and adds a new triangle Δ_(i+1),ateach time,to the intermediate near-triangulation G_i previously formed so that one or twoproperly assigned sides on the bounding circuit of G_i is or are coincident with that of Δ_(i+1).Atthe end of this process one gets G.Based on the above combinatorial results,conjectures which are concerned only with theproperties of 4-colorings of circuits and each of which is equivalent to the Four-Color Theoremare given in the present paper.It is also pointed out that an enlightening conjecture of theabove type—a conjecture at the end of a paper by H.Whitney and W.T.Tutte is not trueeven for circiuts of length 4.