Powers of cycle graph which are k-self complementary and k-co-self complementary

E. Sampath Kumar and L. Pushpalatha [4] introduced a generalized version of complement of a graph with respect to a given partition of its vertex set. Let G = (V, E) be a graph and P = {V1, V2,…, Vk} be a partition of V of order k ≥ 1. The k-complement G^P_k of G with respect to P is defined as follows: For all V_i and V_j in P, i 6= j, remove the edges between Vi and Vj, and add the edges which are not in G. Analogues to self complementary graphs, a graph G is k-self complementary (k-s.c.) if (Formula Presented) and is k-co-self complementary (k-co.s.c.) if (Formula Presented) with respect to a partition P of V (G). The mth power of an undirected graph G, denoted by Gm is another graph that has the same set of vertices as that of G, but in which two vertices are adjacent when their distance in G is at most m. In this article, we study powers of cycle graphs which are k-self complementary and k-co-self complementary with respect to a partition P of its vertex set and derive some interesting results. Also, we characterize k-self complementary C²_n and the respective partition P of V (C²_n ). Finally, we prove that none of the C²_n is k-co-self complementary for any partition P of V (C²_n ).