Avoiding and Extending Partial Edge Colorings of Hypercubes

Abstract

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $$\varphi $$ φ of the d-dimensional hypercube $$Q_d$$ Q d , we are interested in whether there is a proper d-edge coloring of $$Q_d$$ Q d that agrees with the coloring $$\varphi $$ φ on every edge that is colored under $$\varphi $$ φ ; or, similarly, if there is a proper d-edge coloring that disagrees with $$\varphi $$ φ on every edge that is colored under $$\varphi $$ φ . In particular, we prove that for any $$d\ge 1$$ d ≥ 1 , if $$\varphi $$ φ is a partial d-edge coloring of $$Q_d$$ Q d , then $$\varphi $$ φ is avoidable if every color appears on at most d/8 edges and the coloring satisfies a relatively mild structural condition, or $$\varphi $$ φ is proper and every color appears on at most $$d-2$$ d - 2 edges. We also show that $$\varphi $$ φ is avoidable if d is divisible by 3 and every color class of $$\varphi $$ φ is an induced matching. Moreover, for all $$1 \le k \le d$$ 1 ≤ k ≤ d , we characterize for which configurations consisting of a partial coloring $$\varphi $$ φ of $$d-k$$ d - k edges and a partial coloring $$\psi $$ ψ of k edges, there is an extension of $$\varphi $$ φ that avoids $$\psi $$ ψ .

Brooks' theorem with forbidden colors

“…The proof is virtually identical to the corresponding result for edge colorings in [7], so we omit it. Note, in particular, that Proposition 3.4 implies that every partial coloring with at most G Δ( ) − 1 colored vertices is avoidable, which is sharp by the example of the join of K d with two vertices u and v, where u and all vertices of K d are colored 1.…”

“…A G Δ( )-coloring of G where every vertex is assigned the same color is not avoidable if χ G G ( ) = Δ( ). We note the following reformulation of a result for edge colorings from [7]. Proposition 3.4.…”

“…A $${\rm{\Delta }}(G)$$‐coloring of $$G$$ where every vertex is assigned the same color is not avoidable if $$\chi (G)={\rm{\Delta }}(G)$$. We note the following reformulation of a result for edge colorings from [7].…”

“…In this short note we consider the similar problem of constructing proper $${\rm{\Delta }}(G)$$‐colorings avoiding certain colors; that is, given a subset $$P\subseteq V(G)$$ of the vertices of a graph $$G$$ where every vertex of $$P$$ is assigned a set of forbidden colors from $$\{1,\ldots ,{\rm{\Delta }}(G)\}$$, we are interested in finding a proper $${\rm{\Delta }}(G)$$‐coloring of $$G$$ respecting these constraints. These type of questions go back to a paper by Häggkvist [11] and although they arise naturally in problems where a coloring is constructed sequentially, it seems that they so far primarily have been studied in the setting of edge colorings, see for example [6, 5, 7, 9] and references therein. In particular, a variant of Vizing's edge coloring theorem with forbidden colors was obtained in [9].…”

“…of the vertices of a graph G where every vertex of P is assigned a set of forbidden colors from G {1, …, Δ( )}, we are interested in finding a proper G Δ( )-coloring of G respecting these constraints. These type of questions go back to a paper by Häggkvist [11] and although they arise naturally in problems where a coloring is constructed sequentially, it seems that they so far primarily have been studied in the setting of edge colorings, see for example [6,5,7,9] and references therein. In particular, a variant of Vizing's edge coloring theorem with forbidden colors was obtained in [9].…”