Introduction Soft drinks are complex mixtures containing a variety of substances such as coloring compounds, flavoring agents, acidifiers, sweeteners, preservatives, and caffeine. Acidulants reduce the soft drink’s pH and thereby assist in beverage preservation for long-term storage. Acidulants can also be used as chelating agents, buffers, coagulators, and flavoring agents. In the latter role,...

the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...

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...

Graph coloring is a fundamental graph problem that is widely applied in a variety of applications. The aim of graph coloring is to minimize the number of colors used to color the vertices in a graph such that no two incident vertices have the same color. Existing solutions for graph coloring mainly focus on computing a good coloring for a static graph. However, since many real-world graphs are ...

We consider the question of computing the strong edge coloring, square graph coloring, and their generalization to coloring the k power of graphs. These problems have long been studied in discrete mathematics, and their “chaotic” behavior makes them interesting from an approximation algorithm perspective: For k = 1, it is well-known that vertex coloring is “hard” and edge coloring is “easy” in ...

This paper is concerned with algorithms and complexity results for defective coloring, where a defective (k; d)-coloring is a k coloring of the vertices of a graph such that each vertex is adjacent to at most d-self-colored neighbors. First, (2; d) coloring is shown NP-complete for d 1, even for planar graphs, and (3; 1) coloring is also shown NP-complete for planar graphs (while there exists a...

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding a...

This article studies a degree-bounded generalization of independent sets called co-k-plexes. Constant factor approximation algorithms are developed for the maximum co-k-plex problem on unit-disk graphs. The related problem of minimum co-k-plex coloring that generalizes classical vertex coloring is also studied in the context of unit-disk graphs. We extend several classical approximation results...

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 ...