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An acyclic colouring of a graph is a proper colouring of the graph, for which every cycle uses at least three colours. In this paper, we describe a method for constructing all 4-connected acyclically 4-colourable planar triangulations that have exactly four odd-vertices, except the ones that contain no adjacent odd-vertices. Unlike previous operations, our method successfully establishes a connection with (acyclic) 4-colourings. Moreover, we discuss a special class of graphs, called diamond triangulations, and give a necessary and sufficient condition for a diamond triangulation to be acyclically 4-colourable. [ABSTRACT FROM AUTHOR]

A graphGis uniquelyk-colourable if the chromatic number ofGiskandGhas only onek-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 graphGhas at most 4 triangles thenGhas two adjacent triangles. Furthermore, for any, we construct a uniquely 3-colourable planar graph withktriangles and without adjacent triangles. [ABSTRACT FROM PUBLISHER]

A local colouring of a graphGis a functionc: V(G)→ℕ such that for eachS⊆V(G), 2≤|S|≤3, there existu, v∈Swith |c(u)−c(v)| at least the number of edges in the subgraph induced byS. The maximum colour assigned bycis the value χℓ(c) ofc, and the local chromatic number ofGis χℓ(G)=min {χℓ(c):cis a local colouring ofG}. In this note the local chromatic number is determined for Cartesian productsG □ H, whereGandGHare 3-colourable graphs. This result in part corrects an error from Omoomi and Pourmiri [On the local colourings of graphs, Ars Combin. 86 (2008), pp. 147–159]. It is also proved that ifGandHare graphs such that χ(G)≤⌊ χℓ(H)/2 ⌋, then χℓ(G □ H)≤χℓ(H)+1. [ABSTRACT FROM AUTHOR]