Extending partial edge colorings of iterated cartesian products of cycles and paths

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444]

Edge Precoloring Extension of Trees II

We consider the problem of extending and avoiding partial edge colorings of trees; that is, given a partial edge coloring phi of a tree T we are interested in whether there is a proper Delta(T )-edge coloring of T that agrees with the coloring phi on every edge that is colored under phi; or, similarly, if there is a proper Delta(T )-edge coloring that disagrees with phi on every edge that is colored under phi. We characterize which partial edge colorings with at most Delta(T ) + 1 precolored edges in a tree T are extendable, thereby proving an analogue of a result by Andersen for Latin squares. Furthermore we obtain some "mixed" results on extending a partial edge coloring subject to the condition that the extension should avoid a given partial edge coloring; in particular, for all 0 &amp;lt;= k &amp;lt;= Delta(T ), we characterize for which configurations consisting of a partial coloring phi of Delta(T ) - k edges and a partial coloring psi of k + 1 edges of a tree T, there is an extension of phi that avoids psi.Funding Agencies|Swedish Research Council [2017-05077]; International Science Program in Uppsala, Sweden</p