Contractible Non-edges in 3-Connected Infinite Graphs

In this paper, we prove that for any 3-connected finite graph of order n(n≥6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n\ge 6)$$\end{document}, the number of contractible non-edges is at most n(n-5)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n(n-5)}{2}$$\end{document}. All the extremal graphs with at least seven vertices are characterized to be 4-connected 4-regular. By generalizing a result of Kriesell (J Comb Theory Ser B 74:192–201, 1998), we also characterize all 3-connected graphs (finite or infinite) that does not contain any contractible non-edges. In particular, every non-complete 3-connected infinite graph contains a contractible non-edge.