On regular factors in regular graphs with small radius (Q1422147)

Let \(G\) be a \(d\)-regular graph. Obviously \(G\) can only have a \(k\)-factor if \(k| V(G)|\) is even and \(k\leq d\), so let us assume this condition in the following: From Petersen's well-known theorem there follows that then \(G\) has a \(k\)-factor in the case that \(k\) and \(d\) are both even. In the other three cases, if one of \(k\) or \(d\) is odd, \textit{Th. Niessen} and \textit{B. Randerath} [Discrete Math. 185, 89--103 (1998; Zbl 0958.05114)] showed that \(G\) has a \(k\)-factor if the above obvious condition holds and \(G\) is ``small enough.'' In the present paper, the authors show that \(G\) has a \(k\)-factor (again assuming \(k| V(G)|\) to be even and \(k\leq d\)) if the number of vertices of eccentricity 4 and higher is ``small enough.'' The bounds depend on \(k\) and \(d\) and are different in the three remaining cases for parity of \(k\) and \(d\).