Abstract:We investigate locally n×n grid graphs, that is, graphs in which the neighbourhood of any vertex is the Cartesian product of two complete graphs on n vertices. We consider the subclass of these graphs for which each pair of vertices at distance two is joined by sufficiently many paths of length 2. The number of such paths is known to be at most 2n by previous work of Blokhuis and Brouwer. We show that if each distance two pair is joined by at least n − 1 paths of length 2 then the diameter is bounded by O(log(… Show more
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