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    Distance colouring without one cycle length

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    We consider distance colourings in graphs of maximum degree at most dd and how excluding one fixed cycle length \ell affects the number of colours required as dd\to\infty. For vertex-colouring and t1t\ge 1, if any two distinct vertices connected by a path of at most tt edges are required to be coloured differently, then a reduction by a logarithmic (in dd) factor against the trivial bound O(dt)O(d^t) can be obtained by excluding an odd cycle length 3t\ell \ge 3t if tt is odd or by excluding an even cycle length 2t+2\ell \ge 2t+2. For edge-colouring and t2t\ge 2, if any two distinct edges connected by a path of fewer than tt edges are required to be coloured differently, then excluding an even cycle length 2t\ell \ge 2t is sufficient for a logarithmic factor reduction. For t2t\ge 2, neither of the above statements are possible for other parity combinations of \ell and tt. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur
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