Restricted extension of sparse partial edge colorings of complete graphs

TL;DR

This paper investigates conditions under which a partial edge coloring of a complete graph can be extended to a full proper coloring using only allowed colors, focusing on sparse configurations.

Contribution

It establishes positive results for extending partial edge colorings under specific sparsity constraints on the coloring and color lists.

Findings

Extension is possible under certain sparsity conditions

Conditions on partial coloring and color lists are sufficient

Results apply to complete graphs with restricted colorings

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.