TY - JOUR

T1 - Forbidden subgraphs and the existence of a 2-factor

AU - Aldred, R. E.L.

AU - Fujisawa, Jun

AU - Saito, Akira

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

PY - 2010/7

Y1 - 2010/7

N2 - In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F 2 be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph Gis said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)-F 2 is a finite class. In particular, we investigate whether H must contain a star (i.e. K1,n for some positive integer n). We prove the following. (1) If |H|=1, then H={K1,2}. (2) If |H|=2, then H contains K1,2 or K1,3. (3) If |H|=3, then H contains a star. But no restriction is imposed on the order of the star. (4) Not all of H with |H|=4 contain a star. For |H|≤2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582), and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)-G({K1,t}) is finite for some t≥3.

AB - In this paper, we consider forbidden subgraphs which force the existence of a 2-factor. Let G be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let F 2 be the class of graphs which have a 2-factor. For a set H of connected graphs of order at least three, a graph Gis said to be H-free if no member of H is an induced subgraph of G, and let G(H) denote the class of graphs in G that are H-free. We are interested in sets H such that G(H) is an infinite class while G(H)-F 2 is a finite class. In particular, we investigate whether H must contain a star (i.e. K1,n for some positive integer n). We prove the following. (1) If |H|=1, then H={K1,2}. (2) If |H|=2, then H contains K1,2 or K1,3. (3) If |H|=3, then H contains a star. But no restriction is imposed on the order of the star. (4) Not all of H with |H|=4 contain a star. For |H|≤2, we compare our results with a recent result by Faudree et al. (Discrete Math 308 (2008), 1571-1582), and report a difference in the conclusion when connected graphs are considered as opposed to 2-connected graphs. We also observe a phenomenon in which H does not contain a star but G(H)-G({K1,t}) is finite for some t≥3.

KW - 2-Factor

KW - Forbidden subgraph

KW - Induced subgraph

KW - Star

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U2 - 10.1002/jgt.20454

DO - 10.1002/jgt.20454

M3 - Article

AN - SCOPUS:77954343695

VL - 64

SP - 250

EP - 266

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 3

ER -