Linear Coloring and Linear Graphs

TL;DR

This paper introduces linear coloring for graphs, provides bounds and algorithms for it, and defines new classes of perfect graphs called co-linear and linear graphs, exploring their properties and relations.

Contribution

It extends the concept of linear coloring from simplicial complexes to graphs, proposes a polynomial-time coloring algorithm, and defines new graph classes with specific chromatic properties.

Findings

Provided an upper bound for the chromatic number of any graph.

Developed a polynomial-time linear coloring algorithm.

Characterized linear graphs as those achieving the lower bound of linear chromatic number in all induced subgraphs.

Abstract

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We provide an upper bound for the chromatic number $\chi(G)$, for any graph $G$, and show that $G$ can be linearly colored in polynomial time by proposing a simple linear coloring algorithm. Based on these results, we define a new class of perfect graphs, which we call co-linear graphs, and study their complement graphs, namely linear graphs. The linear coloring of a graph $G$ is a vertex coloring such that two vertices can be assigned the same color, if their corresponding clique sets are associated by the set inclusion relation (a clique set of a vertex $u$ is the set of all maximal cliques containing $u$); the linear chromatic number $\mathcal{\lambda}(G)$ of $G$ is the least integer $k$ for which $G$ admits a linear coloring with $k$ colors. We show that linear graphs are those graphs $G$ for which the linear chromatic number achieves its theoretical lower bound in every induced subgraph of $G$. We prove inclusion relations between these two classes of graphs and other subclasses of chordal and co-chordal graphs, and also study the structure of the forbidden induced subgraphs of the class of linear graphs.