The road coloring problem is an interesing problem proposed over 40 years ago by Adler, Goodwyn and Weiss. Hundreds of mathematicians worked on this problem and failed to make the conjecture a theorem. Finally, in 2007, it was proved by a 60-year-old Israeli mathematician, Avraham Trahtman. It turns out the prove was quite simple after Trahtman partitioned the graph into cycles and trees. The d...

Introduction. A well-known algorithm for coloring the vertices of a graph is the “greedy algorithm”: given a totally ordered set of colors, each vertex of the graph (taken in some order) is colored with the least color not already used to color an adjacent vertex. When applied to a path graph with at least two vertices, the algorithm uses either 2 or 3 colors, depending on the order in which th...

We show complexity results for some generalizations of the graph coloring problem on two classes of perfect graphs, namely clique trees and unit interval graphs. We deal with the μ-coloring problem (upper bounds for the color on each vertex), the precoloring extension problem (a subset of vertices colored beforehand), and a problem generalizing both of them, the (γ, μ)-coloring problem (lower a...

We describe an approach to optimization based on a multiple-restart quasi-Hopfield network where the only problem-specific knowledge is embedded in the energy function that the algorithm tries to minimize. We apply this method to three different variants of the graph coloring problem: the minimum coloring problem, the spanning subgraph k-coloring problem, and the induced subgraph k-coloring pro...

The synchronizing word of deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into deterministic finite automaton possessing a synchronizing word. The road coloring problem is a problem of synchronizing coloring of direc...

Suppose a graph G = (V,E) with one destination node g (the gateway) and a set of source nodes with integer demands de ning the input of the problem. Let Φ stand for the set of all possible ows φ : E → Z+ sending all demands from the sources to the gateway. Then each φ ∈ Φ de nes a multigraph Gφ = (V,E, φ), where φ represents the multiplicity of the edges. In the ow coloring problem, the objecti...

We define a variant of the H-coloring problem where the number of preimages of certain vertices is predetermined as part of the problem input. We consider the decision and the counting version of the problem, namely the restrictive H-coloring and the restrictive #H-coloring problems, and we provide a dichotomy theorem determining the H’s for which the restrictive H-coloring problem is either NP...

Let Γ be directed strongly connected finite graph of uniform outdegree (constant outdegree of any vertex) and let some coloring of edges of Γ turn the graph into deterministic complete automaton. Let the word s be a word in the alphabet of colors (considered also as letters) on the edges of Γ and let Γs be a mapping of vertices Γ. A coloring is called k-synchronizing if for any word t |Γt| ≥ k ...

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V,E) with vertex weights w such that ∑k i=1 maxv∈Ci w(vi) is minimized, where C1, . . . , Ck are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be...

Many classes of graphs where the vertex coloring problem is polynomially solvable are known, the most prominent being the class of perfect graphs. However, the list-coloring problem is NP-complete for many subclasses of perfect graphs. In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs, where the former admits polynomia...