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Mat. Sb., 2011, Volume 202, Number 9, Pages 53–76 (Mi msb7747)  

This article is cited in 6 scientific papers (total in 6 papers)

Framed $4$-graphs: Euler tours, Gauss circuits and rotating circuits

D. P. Il'yutko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider connected finite $4$-valent graphs with the structure of opposite edges at each vertex (framed $4$-graphs). For any of such graphs there exist Euler tours, in travelling along which at each vertex we turn from an edge to a nonopposite one (rotating circuits); and at the same time, it is not true that for any such graph there exists an Euler tour passing from an edge to the opposite one at each vertex (a Gauss circuit). The main result of the work is an explicit formula connecting the adjacency matrices of the Gauss circuit and an arbitrary Euler tour. This formula immediately gives us a criterion for the existence of a Gauss circuit on a given framed $4$-graph. It turns out that the results are also valid for all symmetric matrices (not just for matrices realisable by a chord diagram).
Bibliography: 24 titles.

Keywords: framed $4$-graphs, Euler tour, Gauss circuit, rotating circuit, adjacency matrix.

DOI: https://doi.org/10.4213/sm7747

Full text: PDF file (651 kB)
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English version:
Sbornik: Mathematics, 2011, 202:9, 1303–1326

Bibliographic databases:

UDC: 515.16+519.17
MSC: 05C38
Received: 01.06.2010

Citation: D. P. Il'yutko, “Framed $4$-graphs: Euler tours, Gauss circuits and rotating circuits”, Mat. Sb., 202:9 (2011), 53–76; Sb. Math., 202:9 (2011), 1303–1326

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. D. P. Ilyutko, V. O. Manturov, I. M. Nikonov, “Parity in knot theory and graph-links”, Journal of Mathematical Sciences, 193:6 (2013), 809–965  mathnet  crossref  mathscinet
    2. I. M. Nikonov, “Khovanov homology of graph-links”, Sb. Math., 203:8 (2012), 1196–1210  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. D. P. Ilyutko, V. S. Safina, “Graph-links: nonrealizability, orientation, and Jones polynomial”, Journal of Mathematical Sciences, 214:5 (2016), 632–664  mathnet  crossref
    4. D. P. Ilyutko, V. O. Manturov, “A parity map of framed chord diagrams”, J. Knot Theory Ramifications, 24:13 (2015), 1541006, 15 pp.  crossref  mathscinet  zmath  isi  scopus
    5. Manturov V.O., “Framed 4-valent graph minor theory II: Special minors and new examples”, J. Knot Theory Ramifications, 24:13, SI (2015), 1541004  crossref  mathscinet  zmath  isi  scopus
    6. Biryukov O.N., “Parity Conditions For Realizability of Gauss Diagrams”, J. Knot Theory Ramifications, 28:1 (2019), 1950015  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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