Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

Carl Johan Casselgren,Lan Anh Pham

doi:10.37236/9552

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.

Highlights

An edge precoloring of a graph G is a proper edge coloring of some subset E ⊆ E(G); a t-edge precoloring is such a coloring with t colors
A t-edge precoloring φ is extendable if there is a proper t-edge coloring f such that f (e) = φ(e) for any edge e that is colored under φ; f is called an extension of φ
Questions on extending a partial edge coloring seem to have been first considered for balanced complete bipartite graphs, and these questions are usually referred to as problems on completing partial Latin squares

Summary

Introduction

An edge precoloring (or partial edge coloring) of a graph G is a proper edge coloring of some subset E ⊆ E(G); a t-edge precoloring is such a coloring with t colors. Step III: Let h be the proper m-edge coloring satisfying conditions (a)-(i) of Lemma 4 obtained in the previous step.

Results

Conclusion