Geodesic
A Chvátal-Erdős type theorem for path-connectivity
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1247-1260

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For a graph G, let κ(G) and α(G) be the connectivity and independence number of G, respectively. A well-known theorem of Chvátal and Erdős says that if G is a graph of order n with κ(G) gt;α(G), then G is Hamilton-connected. In this paper, we prove the following Chvátal-Erdős type theorem: if G is a k-connected graph, k≥ 2, of order n with independence number α, then each pair of distinct vertices of G is joined by a Hamiltonian path or a path of length at least (k-1)max{n+α-kα, ⌊n+2α-2k+1α⌋}. Examples show that this result is best possible. We also strength it in terms of subgraphs.
Keywords: connectivity, independence number, Hamilton-connected, Chvátal-Erdős theorem 
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     title = {A {Chv\'atal-Erd\H{o}s} type theorem for path-connectivity},
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Chen, Guantao; Hu, Zhiquan; Wu, Yaping. A Chvátal-Erdős type theorem for path-connectivity. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1247-1260. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a0/