Least Common Multiple of a Cycle and a Star

Abstract

A graph G is decomposable into the subgraphs G₁, G₂,…, G_n of G if no G_i, (i = 1, 2,…, n) has isolated vertices and the edge set E(G) can be partitioned into the subsets E(G₁),E(G₂),…, E(G_n). If G_i ≈ H for every i, we say that G is H-decomposable and we write H∣G. A graph F without isolated vertices is a least common multiple of the graphs G₁ and G₂, if F is a graph of minimum size such that F is both G₁-decomposable and G₂-decomposable. The size (the number of edges) of a least common multiple of two graphs G₁ and G₂ is denoted by lcm (G₁,G₂). G. Chartrand et al [1], found lcm (C_2k,K₁,i) and lcm (C₃,K₁,l). For general odd integer n, they introduced a conjecture. Ping Wang [4] proved the conjecture true when n = 5. In this paper, we show that the conjecture is true for the case when l is an odd integer and (n,l) = 1. When 1 < d < l and n/d · d+1/2 ≥ 2l/d + 1, where d = gcd(n, l), we introduce a new formula and prove it.