Divisible subdivisions

Abstract

We prove that for every graph H of maximum degree at most 3 and for every positive integer q there is a finite f = f (H, q) such that every K f -minor contains a subdivision of H in which every edge is replaced by a path whose length is divisible by q. For the case of cycles we show that for f = O(q log q) every K f -minor contains a cycle of length divisible by q, and observe that this settles a recent problem of Friedman and the second author about cycles in (weakly) expanding graphs.

Zero sum cycles in complete digraphs

“…The motivation of Alon and Krivelevich for studying the function n(Z q ) came from a related problem about the containment of cycles with particular lengths in minors of complete graphs. Our improvement on the upper bound on n(Z q ) also directly improves the corresponding result from [2].…”

“…More generally we prove that for every finite Abelian group A we have n(A) ≤ 8|A|, while if |A| is prime then n(A) ≤ 3 2 |A|. As a corollary we also obtain that every K16q-minor contains a cycle of length divisible by q for every integer q ≥ 2, which improves a result from [2].…”

“…In [2] Alon and Krivelevich proved that n(Z q ) ≤ ⌈2q ln q⌉ for every integer q ≥ 2 and n(Z p ) ≤ 2p − 1 for every prime number p. Here we improve their results and show that n(A) grows at most linearly for every non-trivial finite Abelian group A.…”

“…Here we consider this problem for complete directed graphs, where the desired subgraph is a directed cycle. This scenario was recently also considered by Alon and Krivelevich [2].…”

“…Finally we give the proof of Corollary 1. Even though the argument is identical to the one in in [2], we include it for the reader's convenience.…”