Pelabelan L (2, 1) pada Graf Path Union Jewel dan Jaring Berlipat

Abstract

L(2,1) labeling is defined as a function f:V→{0,1,2,…,k} such that |f(u)-f(v)|≥1 if d(u,v)=2 and |f(u)-f(v)|≥2 if d(u,v)=1. The label k is the span of the L(2,1) labeling. The minimum span of a graph G is denoted by λ_2,1 (G). This research aims to obtain the minimum span of the L(2,1) labeling on the path union of jewel graphs and folded-net graphs. The path union of a connected graph G denoted by P_m (G) for m≥2, where m represents the nuber of copies of the graph G, namely G_1,G_2,…,G_m. It is a path that connects a specific vertex in graph G_i to a specific vertex on the next copy, namely G_(i+1) for i=1,2,…,m-1, which results in a path graph P_m. The graph used in the study are the jewel graph (J_n ) with n≥1, the path union jewel graph (P_m (J_n )) with n≥4, the folded- net graph (L_3,3 (r)) with r≥2, and the folded-net path union graph (P_m (L_3,3 (r))) with r≥4. In this research we obtain the minimum span of jewel graph (J_n ) for n≥1 is λ_2,1 (J_n )=n+4, while the path union of jewel graph (P_m (J_n )) for n≥4 is λ_2,1 (P_m (J_n ))=n+4. Wheres on the folded-net graph (L_3,3 (r)) for r≥2 is λ_2,1 (L_3,3 (r))=r+4. Next, for the path union folded-net path graph (P_m (L_3,3 (r))) for r≥4 is λ_2,1 (P_m (L_3,3 (r)))=r+4.