Disjoint K_{\mathrm{4}}^{-} in claw-free graphs with minimum degree at least five

Abstract

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_1,3. Let K_{\mathrm{4}}^{-} be the graph obtained by removing exactly one edge from K₄ and let k be an integer with k\ge \mathrm{2}. We prove that if G is a claw-free graph of order at least 13k - 12 and with minimum degree at least five, then G contains k vertex-disjoint copies of K_{\mathrm{4}}^{-}. The requirement of number five is necessary.

Key words： Forbidden graph; vertex-disjoint subgraph; minimum degree

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