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Chromatic Excellence in Fuzzy Graphs
Authors: K. M. Dharmalingam and R. Udaya Suriya
Number of views: 677
Let $G$ be a simple fuzzy graph. A family $\Gamma^f = \{\gamma_1,\gamma_2,\ldots,\gamma_k\}$ of fuzzy sets on a set $V$ is called $k$-fuzzy colouring of $V=(V,\sigma,\mu)$ if\\ i) $\cup \Gamma^f = \sigma$,\\ ii) $\gamma_i\cap \gamma_j = \emptyset$,\\ iii) for every strong edge $(x,y)$(i.e., $\mu(xy)>0)$ of $G$ \begin{center}$min\{\gamma_i(x),\gamma_i(y)\} = 0,(1\leq i \leq k)$.\end{center} The minimum number of $k$ for which there exists a $k$-fuzzy colouring is called the fuzzy chromatic number of $G$ denoted as $\chi^f(G)$. Then $\Gamma^f$ is the partition of independent sets of vertices of $G$ in which each sets has the same colour is called the fuzzy chromatic partition. A graph $G$ is called the $\chi^f$-excellent if every vertex of $G$ appears as a singleton in some $\chi^f$-partitions of $G$. This paper aims at the study of the new concept namely Chromatic excellence in fuzzy graphs. Fuzzy corona and fuzzy independent sets is defined and studied. We explain these new concepts through examples.