\({K_5^-}\) -factor in a Graph

Abstract

Let \(G\) be a graph and let \(\delta(G)\) denote the minimum degree of \(G\). Let \(F\) be a given connected graph. Suppose that \(|V(G)|\) is a multiple of \(|V(F)|\). A spanning subgraph of \(G\) is called an \(F\)-factor if its components are all isomorphic to \(F\). In 2002, Kawarabayashi [5] conjectured that if \(G\) is a graph of order \(n\) (\(n \geq 3\)) with \(\delta(G) \geq \frac{\ell^2-3\ell+1}{\ell-2}\), then \(G\) has a \(K_\ell^-\)-factor, where \(K_\ell^-\) is the graph obtained from \(K_\ell\) by deleting just one edge. In this paper, we prove that this conjecture is true when \(\ell = 5\).