an:00519832
Zbl 0789.05038
Xie, Litong
Combinatorial operations on near-triangulations of the plane
ZH
J. Syst. Sci. Math. Sci. 13, No. 4, 323-330 (1993).
00016296
1993
j
05C15 05C99
plane; four-color theorem; combinatorial operations; near-triangulations; circuit; triangle; 4-colorings
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 [\textit{H. Whitney} and \textit{W. T. Tutte}, Util. Math. 2, 241-281 (1972; Zbl 0253.05120)] is not true even for circuits of length 4.
Zbl 0253.05120