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Avoiding and Extending Partial Edge Colorings of Hypercubes by Casselgren, Carl Johan; Johansson, Per; Markström, Klas is a scholarly article available to read on EtoBox.

What is Avoiding and Extending Partial Edge Colorings of Hypercubes about?

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 $d\geq 1$, 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 $\psi$.

Author

Casselgren, Carl Johan; Johansson, Per; Markström, Klas

Published

2021

Language

EN