A graph \(G\) is called \({claw-free}\) if \(G\) has no induced subgraph isomorphic to \(K_{1,3}\). Ando et al. obtained the result: a claw-free graph \(G\) with minimum degree at least \(d\) has a path-factor such that the order of each path is at least \(d+1\); in particular \(G\) has a \(\{P_3, P_4, P_5\}\)-factor whenever \(d \geq 2\). Kawarabayashi et al. proved that every \(2\)-connected cubic graph has a \(\{P_3, P_4\}\)-factor. In this article, we show that if \(G\) is a connected claw-free graph with at least \(6\) vertices and minimum degree at least \(2\), then \(G\) has a \(\{P_3, P_4\}\)-factor. As an immediate consequence, it follows that every claw-free cubic graph (not necessarily connected) has a \(\{P_3, P_4\}\)-factor.
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