Contractible Edges in k-Connected Infinite Graphs

In this paper, we prove that every vertex in a k-connected locally finite graph (k≥2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k\ge 2)$$\end{document} which is triangle-free or has minimum degree greater than 32(k-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{2}(k-1)$$\end{document} is incident to at least two contractible edges. Also, it is shown that every vertex in a k-connected locally finite graph (k≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k\ge 3)$$\end{document} with no adjacent triangles is incident to a contractible edge. By restricting to graphs with large minimum end vertex-degree, we generalize Egawa’s result (Graphs Comb 7:15–21, 1991) and prove that every k-connected locally finite infinite graph such that the minimum degree is at least ⌊5k4⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lfloor \frac{5k}{4}\rfloor $$\end{document} and all ends have vertex-degree greater than k contains a contractible edge. We also generalize Dean’s result (J Comb Theory Ser B 48:1–5, 1990) and prove that for any k-connected locally finite infinite graph G(k≥4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k\ge 4)$$\end{document} with minimum end vertex-degree greater than k which is triangle-free or has minimum degree at least ⌊3k2⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lfloor \frac{3k}{2}\rfloor $$\end{document}, the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of G is topologically 2-connected.