# zbMATH — the first resource for mathematics

Spanning $$k$$-ended trees of a claw-free graph. (English) Zbl 1358.05062
Summary: For a tree $$T$$, a vertex of $$T$$ with degree one is often called a leaf of $$T$$. Let $$k\geq2$$ be an integer. We prove that if a connected claw-free graph $$G$$ satisfies $$\alpha^3(G)\leq k$$, then $$G$$ has a spanning tree having at most $$k$$ leaves, where $$\alpha^3(G)$$ denotes the maximum number of vertices of $$G$$ that are pairwise distance at least three in $$G$$. This result implies a known result proved by M. Kano et al. [Ars Comb. 103, 137–154 (2012; Zbl 1265.05100)] which states that if the minimum degree sum of independent $$k+1$$ vertices of a connected claw-free graph $$G$$ is at least $$|G|-k$$, then $$G$$ has a spanning $$k$$-ended tree. The condition on $$\alpha^3(G)$$ is sharp.
##### MSC:
 05C05 Trees
##### Keywords:
spanning tree; claw-free graph; leaf; $$k$$-ended tree
Full Text: