The existence of a 2-factor in K1, n-free graphs with large connectivity and large edge-connectivity

Abstract

In this article, we study the existence of a 2-factor in a K1, n-free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n⩾4, an (n−1)-connected K1, n-free graph of even order has a 1-factor. On the other hand, for every pair of integers m and n with m⩾n⩾4, there exist infinitely many (n−2)-connected K1, n-free graphs of even order and minimum degree at least m which have no 1-factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1-factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n⩾3, every K1, n-free graph of minimum degree at least 2n−2 has a 2-factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge-connectivity and minimum degree to the existence of a 2-factor in a K1, n-free graph are more complicated than those to the existence of a 1-factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K1, n-free graph with a given connectivity or edge-connectivity to have a 2-factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77-89, 2011 © 2011 Wiley Periodicals, Inc.