论文标题
在足够的条件下，图表具有跨度$ k-$结束的树
Progress on sufficient conditions for a graph to have a spanning $k-$ended tree
论文作者
论文摘要
1998年，Broersma和Tuinstra [J。 Graph Doemens \ textbf {29}（1998），227-237]证明，如果$ g $是一个连接的图形，满足$σ_2（g）\ geq | g | -k | -k+1 $，则$ g $具有跨度$ k-$ the。他们还举了一个例子，以表明条件“ $σ_2（g）\ geq | g | -k+1 $”是尖锐的。在本文中，我们引入了这个结果的新进展。令$ k_ {m，m+k} $为一个完整的两部分图，带有两部分$ v（k_ {m，m+k}）= a \ cup b，| a | = m，| = m，| b | = m+k。$ h $表示为从$ k_ {m，m+k} $添加$ we（或添加$ we）的图表$ g $是一个连接的图形，满足$σ_2（g）\ geq | g | -k $，然后$ g $具有跨度$ k-$结束的树，除了$ g $的情况外，$ g $是图形$h。$作为我们的主要结果的$ h. $，是我们的主要条件，可以给出一些图形的足够条件。
In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237] proved that if $G$ is a connected graph satisfying $σ_2(G) \geq |G|-k+1$ then $G$ has a spanning $k-$ended tree. They also gave an example to show that the condition "$σ_2(G) \geq |G|-k+1$" is sharp. In this paper, we introduce a new progress for this result. Let $K_{m,m+k}$ be a complete bipartite graph with bipartition $V(K_{m,m+k})=A\cup B, |A|=m, |B|=m+k.$ Denote by $H$ to be the graph obtained from $K_{m,m+k}$ by adding (or no adding) some edges with two end vertices in $A.$ We prove that if $G$ is a connected graph satisfying $σ_2(G) \geq |G|-k$ then $G$ has a spanning $k-$ended tree except for the case $G$ is isomorphic to a graph $H.$ As a corollary of our main result, a sufficient condition for a graph to have a few branch vertices is given.
