, 4? ? 1})-edge-colouring ?. Now, by Lemma 3.2, for every colour i assigned by ?, the graph M i is 1-planar, and thus admits a proper 6-vertex-colouring ? i . Every two adjacent edges of G are assigned different colours by ?, while, for every two edges at distance 2 being assigned colour i by ?, the two corresponding vertices in M i receive different colours by ? i . Thus ? and the ? i 's yield a strong
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