The k-Restricted Edge Connectivity of Balanced Bipartite Graphs

For a connected graph G = (V, E), an edge set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S\subset E}$$\end{document} is called a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U_1,U_2\subset V(G)}$$\end{document}, denote the set of edges of G with one end in U1 and the other in U2 by [U1, U2]. Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\xi_k(G)=\min\{|[U,V(G){\setminus} U]|: U}$$\end{document} is a vertex subset of order k of G and the subgraph induced by U is connected}. A graph G is said to be λk-optimal if λk(G) = ξk(G). A graph is said to be super-λk if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. In this paper, we present some degree-sum conditions for balanced bipartite graphs to be λk-optimal or super-λk. Moreover, we demonstrate that our results are best possible.