论文标题
$ k $叶连接的图表的足够条件的改善
An improvement of sufficient condition for $k$-leaf-connected graphs
论文作者
论文摘要
对于Integer $ k \ geq2，$ a Graph $ g $称为$ k $ - 叶 - 与$ | v（g）| \ geq k+1 $相连，并给定任何子集$ s \ subseteq v（g）带有$ | s | s | = k = k = k，$ $ g $始终具有$ t $ s $ s $ s $ s $ a $ a $ a a $ a s $而且只有与汉密尔顿相关。在本文中，我们根据大小提供了最佳条件，以确保图形为$ k $ - 叶连接，这不仅改善了Gurgel和Wakabayashi的结果（在$ K $ lileaf-Leaf-Leaf的图形上，J. Combin。理论ser。 B 41（1986）1-16]和AO，Liu，Yuan和Li [改善了$ K $ - 叶与连接的图形的足够条件，离散应用。数学。 314（2022）17-30]，但也扩展了Xu，Zhai和Wang的结果[汉密尔顿连接图的光谱条件的改善，线性多线性代数，2021]。我们的关键方法是表明，如果$（N+K-1）$ - 封闭的非 - $ K $连接的图形必须包含一个大集团，如果其尺寸足够大。作为应用程序，还提供了足够的条件，即以$ k $ $ g $或其补充的（无标志性的laplacian）光谱半径为$ k $ lapeaf连接。
For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is $2$-leaf-connected if and only if it is Hamilton-connected. In this paper, we present a best possible condition based upon the size to guarantee a graph to be $k$-leaf-connected, which not only improves the results of Gurgel and Wakabayashi [On $k$-leaf-connected graphs, J. Combin. Theory Ser. B 41 (1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for $k$-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also extends the result of Xu, Zhai and Wang [An improvement of spectral conditions for Hamilton-connected graphs, Linear Multilinear Algebra, 2021]. Our key approach is showing that an $(n+k-1)$-closed non-$k$-leaf-connected graph must contain a large clique if its size is large enough. As applications, sufficient conditions for a graph to be $k$-leaf-connected in terms of the (signless Laplacian) spectral radius of $G$ or its complement are also presented.
