Abstract
Let \(C \subseteq [r]^m\) be a code such that any two words of C have Hamming distance at least t. It is not difficult to see that determining a code C with the maximum number of words is equivalent to finding the largest n such that there is an r-edge-coloring of \(K_{m, n}\) with the property that any pair of vertices in the class of size n has at least t alternating paths (with adjacent edges having different colors) of length 2. In this paper we consider a more general problem from a slightly different direction. We are interested in finding maximum t such that there is an r-edge-coloring of \(K_{m,n}\) such that any pair of vertices in class of size n is connected by t internally disjoint and alternating paths of length 2k. We also study a related problem in which we drop the assumption that paths are internally disjoint. Finally, we introduce a new concept, which we call alternating connectivity. Our proofs make use of random colorings combined with some integer programs.
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A. Dudek was supported in part by Simons Foundation Grant #244712 and by the National Security Agency under Grant Number H98230-15-1-0172. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon.
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Bennett, P., Dudek, A. & Laforge, E. On the Number of Alternating Paths in Bipartite Complete Graphs. Graphs and Combinatorics 33, 307–320 (2017). https://doi.org/10.1007/s00373-017-1771-x
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DOI: https://doi.org/10.1007/s00373-017-1771-x