Bipartite Subgraphs of Graphs with Maximum Degree Three

We prove that for a connected graph G with maximum degree 3 there exists a bipartite subgraph of G containing almost \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} of the edges of G. Furthermore, we completely characterize the set of all extremal graphs, i.e. all connected graphs G=(V, E) with maximum degree 3 for which no bipartite subgraph has more than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} of the edges; |E| denotes the cardinality of E. For 2-edge-connected graphs there are two kinds of extremal graphs which realize the lower bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}.