An independent set S of a connected graph G is called a frame if G-S is connected. If |S|=k, then... more An independent set S of a connected graph G is called a frame if G-S is connected. If |S|=k, then S is called a k-frame. We prove the following theorem. Let k≥2 be an integer, G be a connected graph with V(G)={v 1 ,v 2 ,⋯,v n }, and deg G (u) denote the degree of a vertex u. Suppose that for every 3-frame S={v a ,v b ,v c } such that 1≤a<b<c≤n, deg G (v a )≤a, deg G (v b )≤b-1 and deg G (v c )≤c-2, it holds that deg G (v a )+deg G (v b )+deg G (v c )-|N G (v a )∩N G (v b )∩N G (v c )|≥|G|-k+1. Then G has a spanning tree with at most k-leaves. Moreover, the condition is sharp. This theorem is a generalization of the results of E. Flandrin et al. [Discrete Math. 90, No. 1, 41–52 (1991; Zbl 0746.05038)] and of A. Kyaw [Australas. J. Comb. 37, 3–10 (2007; Zbl 1120.05019)] for traceability.
For a graph H and an integer k≥2 let σ k (H) denote the minimum degree sum of k independent verti... more For a graph H and an integer k≥2 let σ k (H) denote the minimum degree sum of k independent vertices of H. We prove that if a connected claw-free graph G satisfies σ k+1 (G)≥|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.
An independent set S of a connected graph G is called a frame if G-S is connected. If |S|=k, then... more An independent set S of a connected graph G is called a frame if G-S is connected. If |S|=k, then S is called a k-frame. We prove the following theorem. Let k≥2 be an integer, G be a connected graph with V(G)={v 1 ,v 2 ,⋯,v n }, and deg G (u) denote the degree of a vertex u. Suppose that for every 3-frame S={v a ,v b ,v c } such that 1≤a<b<c≤n, deg G (v a )≤a, deg G (v b )≤b-1 and deg G (v c )≤c-2, it holds that deg G (v a )+deg G (v b )+deg G (v c )-|N G (v a )∩N G (v b )∩N G (v c )|≥|G|-k+1. Then G has a spanning tree with at most k-leaves. Moreover, the condition is sharp. This theorem is a generalization of the results of E. Flandrin et al. [Discrete Math. 90, No. 1, 41–52 (1991; Zbl 0746.05038)] and of A. Kyaw [Australas. J. Comb. 37, 3–10 (2007; Zbl 1120.05019)] for traceability.
For a graph H and an integer k≥2 let σ k (H) denote the minimum degree sum of k independent verti... more For a graph H and an integer k≥2 let σ k (H) denote the minimum degree sum of k independent vertices of H. We prove that if a connected claw-free graph G satisfies σ k+1 (G)≥|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.
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