Some counterexamples associated with the three-color problem

The problems under consideration here were united for the first time by the general term “The Three Color Problem” in Ore’s book [7], where the author was interested in the description of planar 3-colorable graphs. Our graph theoretic terminology is that of Harary [4], except that we use vertices and edges instead of points and lines, respectively. The graphs considered here are planar and have no loops or multiple edges. The history of the three-color problem began in 1958 with the paper of Grotzsch [2]. where it was proved that a planar graph is 3-colorable if it has no triangles. Four years later Grtinbaum [33 proved the following generalization of Griitzsch’s result: Every planar graph with fewer than four triangles is 3-colorable. In the same paper Grtinbaum put forth the question which determined the direction of further research. He supposed that if d is the distance of the triangles (i.e.‘, the length of the shortest path joining vertices of different triangles) and d > 1,. then the planar graph is 3-colorable for an arbitrary number t of triangles. Have1 [5] refuted this conjecture by constructing a planar non-3-colorable graph with d = 1 and t = 4 (Fig. 1) and stated the conjecture that if d > 2 for a planar graph then it is 3-colorable. However, in his next paper Have1 [6] refuted the latter conjecture by constructing a planar non-3-colorable graph with d = 2 and t = 6 (Fig. 2) and proposed the more cautions question: Is there an integer n, such that each graph with d > n,, and an arbitrary number of triangles is 3-colorable? Since the last example due to Have1 contains six triangles, Sachs [S, p. 2581 proposed the