## Spanning trees with a bounded number of leaves in a claw-free graph.(English)Zbl 1265.05100

Summary: For a graph $$H$$ and an integer $$k\geq 2$$ let $$\sigma _k(H)$$ denote the minimum degree sum of $$k$$ independent vertices of $$H$$. We prove that if a connected claw-free graph $$G$$ satisfies $$\sigma _{k+1}(G)\geq | G| -k$$, then $$G$$ has a spanning tree with at most $$k$$ leaves. We also show that the bound $$| G| -k$$ is sharp and discuss the maximum degree of the required spanning trees.

### MSC:

 05C05 Trees 05C35 Extremal problems in graph theory

### Keywords:

spanning tree; leaf; degree sum; claw-free graph