Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

Abstract

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

Restricted extension of sparse partial edge colorings of hypercubes

“…Combining the notion of extending a precoloring and avoiding a list assignment, Andren et al [3] proved that every "sparse" partial edge coloring of K n,n can be extended to a proper n-edge coloring avoiding a given list assignment L satisfying certain "sparsity" conditions, provided that no edge e is precolored by a color that appears in L(e); we refer to [3] for the exact definition of "sparse" in this context. An analogous result for complete graphs was recently obtained in [8].…”

Avoiding and Extending Partial Edge Colorings of Hypercubes

“…Combining the notions of extending a precoloring and avoiding a list assignment, Andren et al [3] proved that a ''sparse'' partial edge coloring of K n;n can be extended to a proper n-edge coloring avoiding a given list assignment L satisfying certain ''sparsity'' conditions, provided that no edge e is precolored by a color that appears in L(e); we refer to [3] for the exact definition of ''sparse'' in this context. An analogous result for complete graphs was recently obtained in [8].…”

Brooks' theorem with forbidden colors

“…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].…”