Regular factors and eigenvalues of regular graphs

In 1985, Bollobás, Saito and Wormald characterized all triples ( t , d , k ) such that every t -edge-connected d -regular graph has a k -factor. An interesting research question is to ask when a t -edge-connected d -regular graph has a k -factor, if the triple ( t , d , k ) does not satisfy the characterization. The problem was solved by Niessen and Randerath in 1998 in terms of a condition involving the number of vertices of the graph. s paper, we continue the investigation of the problem from a spectral perspective. We prove that, for a t -edge-connected d -regular graph G with ( t , d , k ) violating the characterization of Bollobás et al., if a certain eigenvalue, whichever depends on ( t , d , k ) , is not too large (also depends on ( t , d , k ) ), then G still has a k -factor. We also provide sufficient eigenvalue conditions for a t -edge-connected d -regular graph to be k -critical and factor-critical, respectively. Our results extend the characterization of Bollobás, Saito and Wormald, the results of Cioabă, Gregory and Haemers, the results of O and Cioabă, and the results of Lu.