Using Coloring Function to Partition Vertices in a Fuzzy Graph

Abstract

Motivated by some problems in real life, we consider the problem how to partition vertices of a fuzzy graph $\xi = (V, \sigma , \mu )$. We define a coloring function (or coloring for short) of a fuzzy graph $\xi$ to be a mapping $f: V \rightarrow R$ such that $|\sigma (v)f(v)-\sigma (u)f(u)|\ge \mu (vu)$ for any $v,u\in V$. If $|\{f(v): v\in V\}|\le |\{g(v): v\in V\}|$ for any coloring g, then f is a minimum coloring and the cardinality $|\{f(v): v\in V\}|$ is called the chromatic number of $\xi$, denoted $\chi (\xi )$.

The topic is interesting because a series of results show that the chromatic number problem for fuzzy graphs is essential a new combinatorial optimization problem different from, but having some relations with, the chromatic number problem for crisp graphs.