Continuous monotonic decomposition of corona product Cn⊙ Km

Let G be a simple graph with an edge set E(G). If G1, G2, G3, … , Gr are connected edge disjoint subgraphs of G with E(G) = E(G1) ∪ E(G2) ∪ E(G3) ∪ … ∪ E(Gr), then G1, G2, G3, … , Gr is a decomposition of G. An (a, d) − Continuous Monotonic Decompositions, or (a, d) − CMD, of G is a decomposition of G into r subgraphs G1, G2, G3, … , Gr such that every Gi is connected and |E(Gi)| = a + (i − 1)d, for every i = 1, 2, 3, . . , r. Many authors have studied decomposition, including (a, d) − CMD, of graphs. In this paper we study (a, d) − CMD of some other class of graphs. Let n and m be positive integers, n ≥ 3. The corona product of a cycle Cn and an empty graph Km¯, denoted Cn⊙ Km¯, is a graph formed from Cn and n copies of Km¯ by joining each ith vertex of Cn, with an edge, to every vertex of the ith copy of Km¯. A caterpillar is a tree in which the removal of all its end vertices results a path. In this paper we find an (a, d) − CMD of Cn⊙ Km¯ into caterpillars.