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- W4308757910 abstract "For integer $kgeq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|geq k+1$ and given any subset $Ssubseteq V(G)$ 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." @default.
- W4308757910 created "2022-11-15" @default.
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- W4308757910 date "2022-11-09" @default.
- W4308757910 modified "2023-10-14" @default.
- W4308757910 title "An improvement of sufficient condition for $k$-leaf-connected graphs" @default.
- W4308757910 doi "https://doi.org/10.48550/arxiv.2211.04778" @default.
- W4308757910 hasPublicationYear "2022" @default.
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