An Algorithm for Odd Graceful Labeling of the Union of Paths and Cycles

Abstract: In 1991, Gnanajothi [4] proved that the path graph P_n with n vertex and n-1 edge is odd graceful, and the cycle graph C_m with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph C_m $\cup$ P_m when m = 4, 6,8,10 and then we proved that the graph C_ $\cup$ P_n is odd graceful if m is even. Finally, we described an algorithm to label the vertices and the edges of the vertex set V(C_m $\cup$ P_n) and the edge set E(C_m $\cup$ P_n).