Hamiltonian Paths in Connected Claw-Free Graphs

Let \(G\) be a connected claw-free graph of order \(n\). If \(G \not\in M\) and the minimum degree of \(G\) is at least \(\frac{n}{4}\), then \(G\) is traceable.

Here, \(M\) is a set of graphs such that each element in \(M\) can be decomposed into three disjoint subgraphs \(G_1\), \(G_2\), \(G_3\) and \(E_G(G_i, G_j) = u_iu_j\), here \(1 \leq i, j \leq 3\) and \(u_i \in G_i\), \(1\leq i \leq 3\).