Graph colorings and digraph subdivisions

Abstract

This paper presents our studies on three vertex coloring problems on graphs and on a problem concerning subdivision of digraphs. Given an arbitrarily colored graph G, the convex recoloring problem consists in finding a (re)coloring that minimizes the number of color changes and such that each color class induces a connected subgraph of G. This problem is motivated by its application in the study of phylogenetic trees in Bioinformatics. In the kfold coloring problem one wishes to cover the vertices of a graph by a minimum number of stable sets in such a way that every vertex is covered by at least k (possibly identical) sets. The proper orientation problem consists in orienting the edges of a graph so that adjacent vertices have different in-degrees and the maximum in-degree is minimized. Our contributions in these problems are in terms of algorithms, hardness, and polyhedral studies. Finally, we investigate a long-standing conjecture of Mader on subdivision of digraphs: for every acyclic digraph H, there exists an integer f (H) such that every digraph with minimum out-degree at least f (H) contains a subdivision of H as a subdigraph. We give evidences for this conjecture by proving it holds for classes of acyclic digraphs.

A heuristic for the convex recoloring problem in graphs

“…Chopra et al (2016) presented a column generation approach to the problem, solving the same instances when the number of colors is large. Moura (2017) also suggested a column generation approach. The best approximation factor known for the CTR problem is 2 + , presented by Bar-Yehuda et al (2008).…”