The Strong Colors of Flowers - The Structure of Graphs with Chordal Squares

TL;DR

This paper investigates the structure of graphs whose squares are chordal, introducing concepts like flowers and sprouts to characterize these graphs and exploring their implications for coloring problems.

Contribution

It introduces the concepts of flowers, sprouts, and the power of chordality to analyze graphs with chordal squares and line graph squares, advancing understanding of graph coloring complexities.

Findings

Graphs with chordal squares are characterized by the absence of flowers.

The concept of the power of chordality helps in solving coloring NP-hard problems.

Connections between edge and vertex coloring are explored.

Abstract

A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a more general coloring concept can be defined: The strong coloring of a graph. So a k-strong coloring is a coloring where two vertices may not have the same color if their distance to each other is at most k. The 2-strong coloring of the line graph is known as the strong edge coloring. Coloring the kth power G^k of a graph G is the same as finding a k-strong coloring of G itself. In order to obtain a graph class on which the 2-strong coloring becomes efficiently solvable we are looking for a structure that produces induced cycles in the square of G, so that by excluding this structure we obtain a graph class with chordal squares, where a chordal graph is a graph without any induced cycles of length at least 4. Such a structure is called a flower. Another structure will be found and explained, which is responsible for flowers to appear in the line graph of G: The sprouts. With this graphs with chordal line graph squares are described as well. Some attempts in generalizing those structures to obtain perfect graph squares are being made and the general concept of chordal graph powers, i.e. the existence of a smallest power for which a graph becomes chordal, the power of chordality is introduced in order to solve some coloring related NP-hard problems on graphs with parameterized algorithms. Some connections to the famous parameter treewidth arise alongside with some deeper connections between edge and vertex coloring.