TY - JOUR

T1 - The existence of a 2-factor in K1, n-free graphs with large connectivity and large edge-connectivity

AU - Aldred, R. E.L.

AU - Egawa, Yoshimi

AU - Fujisawa, Jun

AU - Ota, Katsuhiro

AU - Saito, Akira

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2011/9

Y1 - 2011/9

N2 - 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.

AB - 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.

KW - (edge-)connectivity

KW - 2-factor

KW - minimum degree

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U2 - 10.1002/jgt.20541

DO - 10.1002/jgt.20541

M3 - Article

AN - SCOPUS:79960966042

VL - 68

SP - 77

EP - 89

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -