The Independence Number Conditions for 2-Factors of a Claw-Free Graph

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number (Formula presented.) is Hamiltonian with one exception of family of graphs. If a nontrivial path contains only internal vertices of degree two and end vertices of degree not two, then we call it a branch. A set S of branches of a graph G is called a branch cut if we delete all edges and internal vertices of branches of S leading to more components than G. We use a branch bond to denote a minimal branch cut. If a branch-bond has an odd number of branches, then it is called odd. In this paper, we shall characterize all 2-connected claw-free graphs G such that every odd branch-bond of G has an edge branch and such that (Formula presented.) but has no 2-factor. We also consider the same problem for those 2-edge-connected claw-free graphs with (Formula presented.).