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- W4297837597 abstract "We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $varphi$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge coloring of $Q_d$ that agrees with the coloring $varphi$ on every edge that is colored under $varphi$; or, similarly, if there is a proper $d$-edge coloring that disagrees with $varphi$ on every edge that is colored under $varphi$. In particular, we prove that for any $dgeq 1$, if $varphi$ is a partial $d$-edge coloring of $Q_d$, then $varphi$ is avoidable if every color appears on at most $d/8$ edges and the coloring satisfies a relatively mild structural condition, or $varphi$ is proper and every color appears on at most $d-2$ edges. We also show that the same conclusion holds if $d$ is divisible by $3$ and every color class of $varphi$ is an induced matching. Moreover, for all $1 leq k leq d$, we characterize for which configurations consisting of a partial coloring $varphi$ of $d-k$ edges and a partial coloring $psi$ of $k$ edges, there is an extension of $varphi$ that avoids $psi$." @default.
- W4297837597 created "2022-10-01" @default.
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- W4297837597 date "2021-04-01" @default.
- W4297837597 modified "2023-10-14" @default.
- W4297837597 title "Avoiding and extending partial edge colorings of hypercubes" @default.
- W4297837597 doi "https://doi.org/10.48550/arxiv.2104.00716" @default.
- W4297837597 hasPublicationYear "2021" @default.
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