Regular factors and eigenvalues of regular graphs

In 1985, Bollobas, 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. In this 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 Bollobas 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 Bollobas, Saito and Wormald, the results of Cioaba, Gregory and Haemers, the results of O and Cioaba, and the results of Lu. (C) 2014 Elsevier Ltd. All rights reserved.