Perfect Pseudo-Matchings in Cubic Graphs

Abstract

A perfect pseudo-matchingM in a cubic graph G is a spanning subgraph of G such that every component of M is isomorphic to K2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_2$$\\end{document} or to K1,3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{1,3}$$\\end{document}. In view of snarks G with dominating cycle C, this is a natural generalization of perfect matchings since G\\E(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G \\setminus E(C)$$\\end{document} is a perfect pseudo-matching. Of special interest are such M where G/M is planar because such G have a cycle double cover. We show that various well known classes of snarks contain planarizing perfect pseudo-matchings, and that there are at least as many snarks with planarizing perfect pseudo-matchings as there are cyclically 5-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5-$$\\end{document}edge-connected snarks.