A note on uniquely 3-colourable planar graphs

Abstract

ABSTRACTA graph G is uniquely k-colourable if the chromatic number of G is k and G has only one k-colouring up to permutation of the colours. Aksionov [On uniquely 3-colorable planar graphs, Discrete Math. 20 (1977), pp. 209–216] conjectured that every uniquely 3-colourable planar graph with at least four vertices has two adjacent triangles. However, in the same year, Melnikov and Steinberg [L.S. Mel'nikov and R. Steinberg, One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977), pp. 203–206.] disproved the conjecture by constructing a counterexample. In this paper, we prove that if a uniquely 3-colourable planar graph G has at most 4 triangles then G has two adjacent triangles. Furthermore, for any , we construct a uniquely 3-colourable planar graph with k triangles and without adjacent triangles.