On Quadrilaterals and 4-path in Claw-free Graphs

Let \( G \) be a claw-free graph of order \( 4k \), where \( k \) is a positive integer. In this paper, it is proved that if the degree sum \( d(u) + d(v) \) is at least \( 4k – 2 \) for every pair of nonadjacent vertices \( u, v \in V(G) \), then \( G \) has a spanning subgraph consisting of \( k – 1 \) quadrilaterals and a 4-path such that all of them are vertex-disjoint, unless \( G \) is isomorphic to \( K_{4k_1 + 2} \cup K_{4k_2 + 2} \), or \( K_{4k_1 + 1} \cup K_{4k_2 + 3} \), where \( k_1 \geq 0, k_2 \geq 0, k_1 + k_2 = k – 1 \). We further showed that the requirement about claw-free is indispensable and the degree condition is sharp.