Some properties on fuzzy chromatic number of union of fuzzy graphs through α-cut graphs coloring

We consider a fuzzy graph G ˜ ( V , E ˜ ) with a crisp vertex set V and a fuzzy edge set E ˜ . A fuzzy graph has some α-cut graphs for α ∈ [0, 1] which are crisp graphs. Coloring of a fuzzy graph could be transformed into classical coloring of the α-cut graphs. Let G ˜ 1 ( V 1 , E ˜ 1 ) and G ˜ 2 ( V 2 , E ˜ 2 ) be two fuzzy graphs with fuzzy chromatic numbers χ ˜ 1 ( G ˜ 1 ) = { ( k , v χ ¯ 1 ( k ) ) } and χ ˜ 2 ( G ˜ 2 ) = { ( k , v χ ¯ 2 ( k ) ) } , respectively. Let G ˜ = G ˜ 1 ∪ G ˜ 2 be a union of G ˜ 1 and G ˜ 2 . By coloring of the α-cut graphs, we get the result that membership function of fuzzy chromatic number of the union χ ˜ ( G ˜ ) = { ( k , v χ ¯ ( k ) ) } satisfies a property v χ ¯ ( k ) = max { v χ ¯ 1 ( k ) , v χ ¯ 2 ( k ) } for k ∈ {1, 2, 3, …, | V 1 ⋃ V 2 |}. Furthermore, we also verify the connection between the fuzzy chromatic number χ ˜ ( G ˜ ) and max { χ ˜ 1 , χ ˜ 2 } through a defuzzification dependent at a decision level.