BILANGAN KROMATIK UNTUK GRAF FUZZY LENGKAP DAN GRAF FUZZY BIPARTISI LENGKAP

Abstract

A fuzzy graph G(V,σ,μ) is a graph which consist of a set of fuzzy nodes σ : V ® [0,1] and a set of fuzzy edges µ: E® [0,1], so that μ(uv) ≤ σ (u) Ùσ (v) "u,vÎV. A fuzzy graph  G = (V,σ,μ) is called complete if μ(uv) = min{σ(u),σ (v)} "u,vÎV. A fuzzy graph G = (V,σ,μ) is called a bipartite fuzzy graph if set of nodes V can divide into two disjoint sets V₁ and V₂ so that μ(v₁v₂) = 0 if v₁,v₂ÎV₁ or v₁,v₂ÎV₂.  If on a bipartite fuzzy graph G, satisfy μ(uv) = min{σ (u),σ(v)} for all uÎV₁ and vÎV₂, then G is called a complete bipartite fuzzy graph.  A k-colouring on a fuzzy graph G(V,σ,μ) is a family of fuzzy sets on V:  G = {g₁, g₂, g₃,…, g_k} which satisfy the following conditions : i) Ú G = σ;  ii) g_i Ù g_j = 0, for all pair of nodes u, v, which adjacent in a fuzzy graph G and min{gi(u),gi(v)} = 0 (1 ≤ i ≤ k). The smallest positive integer k on k-colouring of a fuzzy graph G is called chromatic number of G. Then can be proved that the chromatic number of a complete fuzzy graph with n nodes is c_F(K_σ) = n and the chromatic number of a complete bipartite fuzzy graph is c_F(K_σ1,σ2) = 2.