The complexity of the empire colouring problem for linear forests

Let r and s be fixed positive integers. Assume that the n vertices of a planar graph are partitioned into blocks (or empires) each containing exactly r vertices. The ( s , r ) -colouring problem ( s - COLr) asks for a colouring of the vertices of the graph that uses at most s colours, never assigns the same colour to adjacent vertices in different empires and, conversely, assigns the same colour to all vertices in the same empire, disregarding adjacencies. For r = 1 the problem coincides with the classical vertex colouring problem on planar graphs. The generalization for r ≥ 2 was defined by Percy Heawood in 1890 in the same paper in which he refuted a previous “proof” of the famous Four Colour Theorem. ecent paper we have shown that if s ≥ 3 , s - COLr is NP-hard for linear forests if s < r . Furthermore, the hardness extends to s < 6 r − 3 (resp. s < 7 ) when r ≥ 3 (resp. for r = 2 ) for arbitrary planar graphs. For trees, our argument entails a nice dichotomy: s - COLr is NP-hard for s ∈ { 3 , … , 2 r − 1 } and solvable in polynomial time for any other positive value of s . In this paper we argue that linear forests do not make the problem any easier, even for small values of r . We prove that the s - COLr problem is NP-hard for linear forests even if r = 2 and s = 3 , or r = 3 and s ∈ { 3 , 4 } .