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arXiv · 2001.09369

On the number of alternating paths in random graphs

Patrick Bennett·Ryan Cushman·Andrzej Dudek
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Abstract

In the noisy channel model from coding theory, we wish to detect errors introduced during transmission by optimizing various parameters of the code. Bennett, Dudek, and LaForge framed a variation of this problem in the language of alternating paths in edge-colored complete bipartite graphs in 2016. Here, we extend this problem to the random graph $\mathbb{G}(n,p)$. We seek the alternating connectivity, $\kappa_{r,\ell}(G)$, which is the maximum $t$ such that there is an $r$-edge-coloring of $G$ such that any pair of vertices is connected by $t$ internally disjoint and alternating (i.e. no consecutive edges of the same color) paths of length $\ell$. We have three main results about how this parameter behaves in $\mathbb{G}(n,p)$ that basically cover all ranges of $p$: one for paths of length two, one for the dense case, and one for the sparse case. For paths of length two, we found that $\kappa_{r,\ell}(G)$ is essentially the codegree of a pair of vertices. For the dense case when $p$ is constant, we were able to achieve the natural upper bounds of minimum degree (minus some intersection) or the total number of disjoint paths between a pair of vertices. For the sparse case, we were able to find colorings that achieved the natural obstructions of minimum degree or (in a slightly less precise result) the total number of paths of a certain length in a graph. We broke up this sparse case into ranges of $p$ corresponding to when $\mathbb{G}(n,p)$ has diameter $k$ or $k+1$. We close with some remarks about a similar parameter and a generalization to pseudorandom graphs.

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Patrick Bennett, Ryan Cushman, Andrzej Dudek. 2020-01-25. On the number of alternating paths in random graphs. https://arxiv.org/abs/2001.09369

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