Let T be a tree. The set of leaves of Τ is denoted by Leaf(Τ). The subtree Τ—Leaf(Τ) of T is called the stem of Τ. A stem is called a k-ended stem if it has at most k-leaves in it. In this paper, we prove the fo...Let T be a tree. The set of leaves of Τ is denoted by Leaf(Τ). The subtree Τ—Leaf(Τ) of T is called the stem of Τ. A stem is called a k-ended stem if it has at most k-leaves in it. In this paper, we prove the following theorem. Let G be a connected graph and k≥2 be an integer. Let u and ν be a pair of nonadjacent vertices in G. Suppose that |NG(u)∪NG(v)|≥|G|-k-1. Then G has a spanning tree with k-ended stem if and only if G+uv has a spanning tree with k-ended stem. Moreover, the condition on |NG(u)∪NG(v)| is sharp.展开更多
Win proved a well-known result that the graph G of connectivity κ(G) withα(G) ≤κ(G) + k-1(k ≥ 2) has a spanning k-ended tree, i.e., a spanning tree with at most k leaves. In this paper, the authors extended the W...Win proved a well-known result that the graph G of connectivity κ(G) withα(G) ≤κ(G) + k-1(k ≥ 2) has a spanning k-ended tree, i.e., a spanning tree with at most k leaves. In this paper, the authors extended the Win theorem in case when κ(G) = 1 to the following: Let G be a simple connected graph of order large enough such that α(G) ≤ k + 1(k ≥ 3) and such that the number of maximum independent sets of cardinality k + 1 is at most n-2k-2. Then G has a spanning k-ended tree.展开更多
文摘Let T be a tree. The set of leaves of Τ is denoted by Leaf(Τ). The subtree Τ—Leaf(Τ) of T is called the stem of Τ. A stem is called a k-ended stem if it has at most k-leaves in it. In this paper, we prove the following theorem. Let G be a connected graph and k≥2 be an integer. Let u and ν be a pair of nonadjacent vertices in G. Suppose that |NG(u)∪NG(v)|≥|G|-k-1. Then G has a spanning tree with k-ended stem if and only if G+uv has a spanning tree with k-ended stem. Moreover, the condition on |NG(u)∪NG(v)| is sharp.
基金supported by the National Natural Science Foundation of China(Nos.11871099,11671037,11801296)the Nature Science Foundation from Qinghai Province(No.2017-ZJ-949Q)
文摘Win proved a well-known result that the graph G of connectivity κ(G) withα(G) ≤κ(G) + k-1(k ≥ 2) has a spanning k-ended tree, i.e., a spanning tree with at most k leaves. In this paper, the authors extended the Win theorem in case when κ(G) = 1 to the following: Let G be a simple connected graph of order large enough such that α(G) ≤ k + 1(k ≥ 3) and such that the number of maximum independent sets of cardinality k + 1 is at most n-2k-2. Then G has a spanning k-ended tree.
