Forbidden subgraphs and the existence of a 2‐factor
Robert E. L. Aldred, Jun Fujisawa, Akira Saito · Journal of Graph Theory · 2009
Abstract In this paper, we consider forbidden subgraphs which force the existence of a 2‐factor. Let \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}$\end{document} be the class of connected graphs of minimum degree at least two and maximum degree at least three, and let \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${{\cal F}_2}$\end{document} be the class of graphs which have a 2‐factor. For a set \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} of connected graphs of order at least three, a graph G is said to be \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} ‐free if no member of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} is an induced subgraph of G, and let \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}({\cal H})$\end{document} denote the class of graphs in \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}$\end{document} that are \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} ‐free. We are interested in sets \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} such that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}({\cal H})$\end{document} is an infinite class while \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}({\cal H})-{\cal F}_2$\end{document} is a finite class. In particular, we investigate whether \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} must contain a star (i.e. K1, n for some positive integer n). We prove the following: If |\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} |=1, then \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} ={K1, 2}. If |\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} |=2, then \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} contains K1, 2 or K1, 3. If |\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} |=3, then \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} contains a star. But no restriction is imposed on the order of the star. Not all of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} with |\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} |=4 contain a star. For |\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} |⩽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 \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} does not contain a star but \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}$\end{document} (\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal H}$\end{document} )−\documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\cal G}$\end{document} ({K1, t}) is finite for some t⩾3. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 250–266, 2010