an:06693852
Zbl 1358.05062
Yan, Zheng; Tsugaki, Masao
Spanning \(k\)-ended trees of a claw-free graph
EN
Adv. Appl. Discrete Math. 17, No. 4, 453-459 (2016).
0974-1658
2016
j
05C05
spanning tree; claw-free graph; leaf; \(k\)-ended tree
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 \textit{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.
1265.05100