Abstract
AbstractThe existence of a function α(k) (where k is a natural number) is established such that the vertex set of any graph G of minimum degree at least α(k) has a decomposition A ∪ B ∪ C such that G(A) has minimum degree at least k, each vertex of A is joined to at least k vertices of B, and no two vertices of B are separated by fewer than k vertices in G(G ∪ C). This is applied to prove the existence of subdivisions of complete bipartite graphs (complete graphs) with prescribed path lengths modulo k in graphs of sufficiently high minimum degree (chromatic number) and path systems with prescribed ends and prescribed lengths modulo k in graphs of sufficiently high connectivity.
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