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

R. E.L. Aldred, Yoshimi Egawa, Jun Fujisawa, Katsuhiro Ota, Akira Saito

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Abstract

In this article, we study the existence of a 2-factor in a K 1, 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 K 1, n-free graph with a given connectivity or edge-connectivity to have a 2-factor.

Original languageEnglish
Pages (from-to)77-89
Number of pages13
JournalJournal of Graph Theory
Volume68
Issue number1
DOIs
Publication statusPublished - 2011 Sep 1

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Keywords

  • (edge-)connectivity
  • 2-factor
  • K-free graph
  • minimum degree

ASJC Scopus subject areas

  • Geometry and Topology

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