On a Sharp Degree Sum Condition for Disjoint Chorded Cycles in Graphs

Let r and s be nonnegative integers, and let G be a graph of order at least 3r + 4s. In Bialostocki et al. (Discrete Math 308: 5886-5890, 2008), conjectured that if the minimum degree of G is at least 2r + 3s, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles, and they showed that the conjecture is true for r = 0, s = 2 and for s = 1. In this paper, we settle this conjecture completely by proving the following stronger statement; if the minimum degree sum of two nonadjacent vertices is at least 4r + 6s - 1, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles.