likimainväritys

Likimainväritys is a Finnish term that translates to "near coloring" or "approximate coloring." It is a concept in graph theory that deals with assigning colors to the vertices of a graph such that adjacent vertices have colors that are "close" to each other, rather than strictly different. This is a relaxation of the standard graph coloring problem.

In traditional graph coloring, adjacent vertices must receive distinct colors. Likimainväritys introduces a notion of distance between colors. If we consider the colors to be integers, then two adjacent vertices $u$ and $v$ might be allowed to have colors $c(u)$ and $c(v)$ such that $|c(u) - c(v)| \le k$ for some non-negative integer $k$. The smallest integer $k$ for which a graph can be properly likimainväritys is called its likimainväritys number.

The study of likimainväritys is motivated by real-world applications where perfect separation of "colors" (e.g., frequencies in wireless communication, or time slots in scheduling) is not always feasible or necessary. Allowing for some overlap or proximity can lead to more efficient or practical solutions. For instance, in frequency assignment, it might be acceptable for adjacent transmitters to use frequencies that are close, as long as the interference is below a certain threshold.

The likimainväritys number of a graph is related to its chromatic number, which is the minimum number of colors needed for a standard vertex coloring. If $k=0$, likimainväritys is equivalent to standard graph coloring. As $k$ increases, the number of colors required generally decreases. Research in this area focuses on determining the likimainväritys number for various classes of graphs and understanding its properties.