Combinatorial refinement on circulant graphs

Abstract

The combinatorial refinement techniques have proven to be an efficientapproach to isomorphism testing for particular classes of graphs.If the number of refinement rounds is small, this putsthe corresponding isomorphism problem in a low-complexity class.We investigate the round complexity of the two-dimensional Weisfeiler--Leman algorithmon circulant graphs, i.e., on Cayley graphs of the cyclic group Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{Z}_n$$\\end{document}, and prove that the number of rounds until stabilization is bounded by O(d(n)logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(d(n)\\log n)$$\\end{document},where d(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(n)$$\\end{document} is the number of divisors of n. As a particular consequence, isomorphism can be tested in NC for connected circulant graphs of order pℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^\\ell$$\\end{document} with p an odd prime, ℓ>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell>3$$\\end{document} and vertex degree Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta$$\\end{document} smaller than p.We also show that the color refinement method (also known as the one-dimensional Weisfeiler--Leman algorithm)computes a canonical labeling for every non-trivial circulant graph with a prime number of verticesafter individualization of two appropriately chosen vertices.Thus, the canonical labeling problem for this class of graphs has at most the samecomplexity as color refinement, which results in a time bound of O(Δnlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(\\Delta \\, n\\log n)$$\\end{document}.Moreover, this provides a first example wherea sophisticated approach to isomorphism testing put forward by Tinhofer has a real practical meaning.