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We show that a graph of girth greater than 6 log k + 3 and minimum degree at least 3 has a minor of minimum degree greater than k. This is best possible up to a factor of at most 9/4. As a corollary, every graph of girth at least 6 log r + 3 log log r + c and minimum degree at least 3 has a Kr minor.

For integers m; d; D with m 3; d 2; and D 2, let T(m) be a 2{dimensional quadratic toroidal grid with side length m and let B(d;D) be the base d, dimension D de Bruijn graph; assume that jT(m)j = jB(d;D)j. The starting point for our investigations is the observation that, for m; D even, embeddings f : T(m) ! B(d; D) with load 1, expansion 1, and dilation… (More)

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