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

Abstract

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,...,V_k} 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 ≠ j, remove the edges between V_i and V_j , 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 G^P_k ≅ G and is k-co-self complementary (k-co.s.c.) if G^P_k ≅ Ġ with respect to a partition P of V (G). The m^th power of an undirected graph G, denoted by G^m 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).