Colouring planar graphs with a precoloured induced cycle
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Let C be a cycle and f : V(C) →{c_1,c_2,…,c_k} a proper k-colouring of C for some k ≥ 4. We say the colouring f is safe if for any planar graph G in which C is an induced cycle, there exists a proper k-colouring f' of G such that f'(v) = f(v) for all v ∈ V(C). The only safe 4-colouring is any proper colouring of a triangle. We give a simple necessary condition for a k-colouring of a cycle to be safe and conjecture that it is sufficient for all k ≥ 4. The sufficiency for k=4 follows from the four colour theorem and we prove it for k = 5, independent of the four colour theorem. We show that a stronger condition is sufficient for all k ≥ 4. As a consequence, it follows that any proper k-colouring of a cycle that uses at most k-3 distinct colours is safe. Also, any proper k-colouring of a cycle of length at most 2k-5 that uses at most k-1 distinct colours is safe.
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