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arXiv · 2607.08062

Extremal problems on disjoint path covers of graphs

Shujie Chen·Tao Tian
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Abstract

In 1962, Erdős characterized the maximum size of nonhamiltonian graphs of order $n$ with minimum degree at least $k$. Later, Ning and Peng [Combin. Probab. Comput. 29 (2020) 128-136] extended Erdős's results to the clique condition and provided the maximum clique number for nonhamiltonian graphs of order $n$ with minimum degree at least $k$. Recently, Zhang [European J. Combin. 112 (2023) 103728] determined the maximum number of $s$-cliques in nonhamiltonian graphs with prescribed order and minimum degree. A natural extension is to characterize the maximum number of $s$-cliques under other graph properties. Notably, disjoint path cover problems are closely related to Hamiltonicity. In this paper, we generalize results on Hamiltonicity and establish sufficient conditions for a graph to possess one-to-one, one-to-many and many-to-many $t$-disjoint path covers in terms of the number of cliques and the $α$-spectral radius, respectively. Furthermore, we characterize the extremal graphs that attain these bounds respectively.

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Shujie Chen, Tao Tian. 2026-07-09. Extremal problems on disjoint path covers of graphs. https://arxiv.org/abs/2607.08062

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