For positive integers, , , and , Bollobás, Saito, and Wormald proved some sufficient conditions for an ‐edge‐connected ‐regular graph to have a ‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected ‐regular graph to have a ‐factor in 2010. Gu found an upper bound for certain eigenvalues in an ‐edge‐connected ‐regular graph to have a ‐factor in 2014. For positive integers , an even (or odd) ‐factor of a graph is a spanning subgraph such that for each vertex , is even (or odd) and . In this paper, we prove upper bounds (in terms of and ) for certain eigenvalues (in terms of and ) in an ‐edge‐connected ‐regular graph to guarantee the existence of an even ‐factor or an odd ‐factor. This result extends the one of Bollbás, Saito, and Wormald, the one of Lu, and the one of Gu.
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