On 2-factors with a specified number of components in line graphs

Abstract

Kaiser and Vr\’{a}na [European J. Combin. 33 (2012) 924--947] showed that every $5$-connected line graph of minimum degree at least $6$ is hamiltonian, which gives a partial solution to Thomassen's Conjecture on hamiltonicity of line graphs [J. Graph Theory 10 (1986) 309--324]. In this paper, we prove that every $5$-connected line graph of sufficiently large order compared with a given positive integer $k$ and of minimum degree at least $6$ also has a $2$-factor with exactly $k$ cycles. In order to show this result, we investigate minimum degree conditions for the existence of such a $2$-factor in hamiltonian line graphs.