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<title>LCF notation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">LCF notation</span></span>

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<p>In the <a href="Mathematical" class="mw-redirect" title="Mathematical">mathematical</a> field of <a href="Graph_theory" title="Graph theory">graph theory</a>, <b>LCF notation</b> or <b>LCF code</b> is a notation devised by <a href="Joshua_Lederberg" title="Joshua Lederberg">Joshua Lederberg</a>, and extended by <a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">H. S. M. Coxeter</a> and <a href="Robert_Frucht" title="Robert Frucht">Robert Frucht</a>, for the representation of <a href="Cubic_graph" title="Cubic graph">cubic graphs</a> that contain a <a href="Hamiltonian_path" title="Hamiltonian path">Hamiltonian cycle</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The cycle itself includes two out of the three adjacencies for each <a href="Vertex_(graph_theory)" title="Vertex (graph theory)">vertex</a>, and the LCF notation specifies how far along the cycle each vertex's third neighbor is. A single graph may have multiple different representations in LCF notation.

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<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>

<p>In a Hamiltonian graph, the vertices can be <a href="Circular_layout" title="Circular layout">arranged in a cycle</a>, which accounts for two edges per vertex. The third edge from each vertex can then be described by how many positions clockwise (positive) or counter-clockwise (negative) it leads. The basic form of the LCF notation is just the sequence of these numbers of positions, starting from an arbitrarily chosen vertex and written in square brackets.

The numbers between the brackets are interpreted <a href="Modular_arithmetic" title="Modular arithmetic">modulo</a> <i>N</i>, where <i>N</i> is the number of vertices. Entries congruent modulo <i>N</i> to 0, 1, or <i>N</i> − 1 do not appear in this sequence of numbers,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> because they would correspond either to a <a href="Loop_(graph_theory)" title="Loop (graph theory)">loop</a> or <a href="Multigraph" title="Multigraph">multiple adjacency</a>, neither of which are permitted in simple graphs.

</p><p>Often the pattern repeats, and the number of repetitions can be indicated by a superscript in the notation. For example, the <a href="Nauru_graph" title="Nauru graph">Nauru graph</a>,<sup id="cite_ref-DE1_1-1" class="reference"><a href="#cite_note-DE1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> shown on the right, has four repetitions of the same six offsets, and can be represented by the LCF notation [5, −9, 7, −7, 9, −5]<sup>4</sup>. A single graph may have multiple different LCF notations, depending on the choices of Hamiltonian cycle and starting vertex.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>LCF notation is useful in publishing concise descriptions of Hamiltonian cubic graphs, such as the examples below. In addition, some software packages for manipulating graphs include utilities for creating a graph from its LCF notation.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>If a graph is represented by LCF notation, it is straightforward to test whether the graph is <a href="Bipartite_graph" title="Bipartite graph">bipartite</a>: this is true if and only if all of the offsets in the LCF notation are odd.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<th>Vertices</th>

<th>LCF notation

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<td><a href="Tetrahedron" title="Tetrahedron">Tetrahedral</a> graph</td>

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<td><a href="Water%2C_gas%2C_and_electricity" class="mw-redirect" title="Water, gas, and electricity">Utility graph</a></td>

</td></tr>

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<td><a href="Hypercube_graph" title="Hypercube graph">Cubical graph</a></td>

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<td><a href="Wagner_graph" title="Wagner graph">Wagner graph</a></td>

</td></tr>

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<td><a href="Bidiakis_cube" title="Bidiakis cube">Bidiakis cube</a></td>

<td>[6,4,−4]<sup>4</sup> or [6,−3,3,6,3,−3]<sup>2</sup> or [−3,6,4,−4,6,3,−4,6,−3,3,6,4]

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<td><a href="Franklin_graph" title="Franklin graph">Franklin graph</a></td>

</td></tr>

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<td><a href="Frucht_graph" title="Frucht graph">Frucht graph</a></td>

<td>[−5,−2,−4,2,5,−2,2,5,−2,−5,4,2]

</td></tr>

<tr>

<td><a href="Truncated_tetrahedron" title="Truncated tetrahedron">Truncated tetrahedral</a> graph</td>

</td></tr>

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<td><a href="Heawood_graph" title="Heawood graph">Heawood graph</a></td>

</td></tr>

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<td><a href="M%C3%B6bius%E2%80%93Kantor_graph" title="Möbius–Kantor graph">Möbius–Kantor graph</a></td>

</td></tr>

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<td><a href="Pappus_graph" title="Pappus graph">Pappus graph</a></td>

</td></tr>

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<td>Smallest <a href="Zero-symmetric_graph" title="Zero-symmetric graph">zero-symmetric graph</a><sup id="cite_ref-FOOTNOTECoxeterFruchtPowers1981Fig.&nbsp;1.1,_p.&nbsp;5_7-0" class="reference"><a href="#cite_note-FOOTNOTECoxeterFruchtPowers1981Fig.&nbsp;1.1,_p.&nbsp;5-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></td>

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<td><a href="Desargues_graph" title="Desargues graph">Desargues graph</a></td>

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<td><a href="Dodecahedron" title="Dodecahedron">Dodecahedral</a> graph</td>

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<td><a href="McGee_graph" title="McGee graph">McGee graph</a></td>

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<td><a href="Truncated_cube" title="Truncated cube">Truncated cubical</a> graph</td>

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<td><a href="Truncated_octahedron" title="Truncated octahedron">Truncated octahedral</a> graph</td>

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<td><a href="Nauru_graph" title="Nauru graph">Nauru graph</a></td>

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<td><a href="F26A_graph" title="F26A graph">F26A graph</a></td>

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<td><a href="Tutte%E2%80%93Coxeter_graph" title="Tutte–Coxeter graph">Tutte–Coxeter graph</a></td>

</td></tr>

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<td><a href="Dyck_graph" title="Dyck graph">Dyck graph</a></td>

</td></tr>

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<td><a href="Gray_graph" title="Gray graph">Gray graph</a></td>

</td></tr>

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<td><a href="Truncated_dodecahedron" title="Truncated dodecahedron">Truncated dodecahedral</a> graph</td>

</td></tr>

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<td><a href="Harries_graph" title="Harries graph">Harries graph</a></td>

</td></tr>

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<td><a href="Harries%E2%80%93Wong_graph" title="Harries–Wong graph">Harries–Wong graph</a></td>

<td>[9, 25, 31, −17, 17, 33, 9, −29, −15, −9, 9, 25, −25, 29, 17, −9, 9, −27, 35, −9, 9, −17, 21, 27, −29, −9, −25, 13, 19, −9, −33, −17, 19, −31, 27, 11, −25, 29, −33, 13, −13, 21, −29, −21, 25, 9, −11, −19, 29, 9, −27, −19, −13, −35, −9, 9, 17, 25, −9, 9, 27, −27, −21, 15, −9, 29, −29, 33, −9, −25]

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<td><a href="Balaban_10-cage" title="Balaban 10-cage">Balaban 10-cage</a></td>

<td>[−9, −25, −19, 29, 13, 35, −13, −29, 19, 25, 9, −29, 29, 17, 33, 21, 9,−13, −31, −9, 25, 17, 9, −31, 27, −9, 17, −19, −29, 27, −17, −9, −29, 33, −25,25, −21, 17, −17, 29, 35, −29, 17, −17, 21, −25, 25, −33, 29, 9, 17, −27, 29, 19, −17, 9, −27, 31, −9, −17, −25, 9, 31, 13, −9, −21, −33, −17, −29, 29]

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<td><a href="Foster_graph" title="Foster graph">Foster graph</a></td>

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<td><a href="Biggs%E2%80%93Smith_graph" title="Biggs–Smith graph">Biggs–Smith graph</a></td>

<td>[16, 24, −38, 17, 34, 48, −19, 41, −35, 47, −20, 34, −36, 21, 14, 48, −16, −36, −43, 28, −17, 21, 29, −43, 46, −24, 28, −38, −14, −50, −45, 21, 8, 27, −21, 20, −37, 39, −34, −44, −8, 38, −21, 25, 15, −34, 18, −28, −41, 36, 8, −29, −21, −48, −28, −20, −47, 14, −8, −15, −27, 38, 24, −48, −18, 25, 38, 31, −25, 24, −46, −14, 28, 11, 21, 35, −39, 43, 36, −38, 14, 50, 43, 36, −11, −36, −24, 45, 8, 19, −25, 38, 20, −24, −14, −21, −8, 44, −31, −38, −28, 37]

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<td><a href="Balaban_11-cage" title="Balaban 11-cage">Balaban 11-cage</a></td>

<td>[44, 26, −47, −15, 35, −39, 11, −27, 38, −37, 43, 14, 28, 51, −29, −16, 41, −11, −26, 15, 22, −51, −35, 36, 52, −14, −33, −26, −46, 52, 26, 16, 43, 33, −15, 17, −53, 23, −42, −35, −28, 30, −22, 45, −44, 16, −38, −16, 50, −55, 20, 28, −17, −43, 47, 34, −26, −41, 11, −36, −23, −16, 41, 17, −51, 26, −33, 47, 17, −11, −20, −30, 21, 29, 36, −43, −52, 10, 39, −28, −17, −52, 51, 26, 37, −17, 10, −10, −45, −34, 17, −26, 27, −21, 46, 53, −10, 29, −50, 35, 15, −47, −29, −41, 26, 33, 55, −17, 42, −26, −36, 16]

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<td><a href="Ljubljana_graph" title="Ljubljana graph">Ljubljana graph</a></td>

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<td><a href="Tutte_12-cage" title="Tutte 12-cage">Tutte 12-cage</a></td>

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Extended_LCF_notation">Extended LCF notation</h2></div>

<p>A more complex extended version of LCF notation was provided by Coxeter, Frucht, and Powers in later work.<sup id="cite_ref-FOOTNOTECoxeterFruchtPowers198154_8-0" class="reference"><a href="#cite_note-FOOTNOTECoxeterFruchtPowers198154-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In particular, they introduced an "anti-palindromic" notation: if the second half of the numbers between the square brackets was the reverse of the first half, but with all the signs changed, then it was replaced by a semicolon and a dash. The Nauru graph satisfies this condition with [5, −9, 7, −7, 9, −5]<sup>4</sup>, and so can be written [5, −9, 7; −]<sup>4</sup> in the extended notation.<sup id="cite_ref-FOOTNOTECoxeterFruchtPowers198112_9-0" class="reference"><a href="#cite_note-FOOTNOTECoxeterFruchtPowers198112-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-DE1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-DE1_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DE1_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="David_Eppstein" title="David Eppstein">Eppstein, D.</a>, <a rel="nofollow" class="external text" href="https://11011110.github.io/blog/2007/12/12/many-faces-of.html">The many faces of the Nauru graph</a>, 2007.</span>

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</style><cite id="CITEREFPisanskiServatius2013" class="citation cs2"><a href="Toma%C5%BE_Pisanski" title="Tomaž Pisanski">Pisanski, Tomaž</a>; <a href="Brigitte_Servatius" title="Brigitte Servatius">Servatius, Brigitte</a> (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bnh2zkuTZr4C&pg=PA32">"2.3.2 Cubic graphs and LCF notation"</a>, <i>Configurations from a Graphical Viewpoint</i>, Springer, p. 32, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780817683641</bdi></cite>.</span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFFrucht1976" class="citation cs2">Frucht, R. (1976), "A canonical representation of trivalent Hamiltonian graphs", <i><a href="Journal_of_Graph_Theory" title="Journal of Graph Theory">Journal of Graph Theory</a></i>, <b>1</b> (1): <span class="nowrap">45–</span>60, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fjgt.3190010111">10.1002/jgt.3190010111</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463029">0463029</a></cite>.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFKutnarMarušič2008" class="citation cs2"><a href="Klavdija_Kutnar" title="Klavdija Kutnar">Kutnar, Klavdija</a>; <a href="Dragan_Maru%C5%A1i%C4%8D" title="Dragan Marušič">Marušič, Dragan</a> (2008), "Hamiltonicity of vertex-transitive graphs of order <span class="texhtml">4<i>p</i></span>", <i><a href="European_Journal_of_Combinatorics" title="European Journal of Combinatorics">European Journal of Combinatorics</a></i>, <b>29</b> (2): <span class="nowrap">423–</span>438, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0606585">math/0606585</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ejc.2007.02.002">10.1016/j.ejc.2007.02.002</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2388379">2388379</a></cite>. See Section 2.</span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">e.g. <a rel="nofollow" class="external text" href="http://www.maplesoft.com/support/help/AddOns/view.aspx?path=GraphTheory/SpecialGraphs/LCFGraph">Maple</a>, <a rel="nofollow" class="external text" href="http://networkx.lanl.gov/reference/generated/networkx.LCF_graph.html">NetworkX</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120302195926/http://networkx.lanl.gov/reference/generated/networkx.LCF_graph.html">Archived</a> 2012-03-02 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, <a rel="nofollow" class="external text" href="http://igraph.org/c/doc/igraph-Generators.html#igraph_lcf">igraph</a>, and <a rel="nofollow" class="external text" href="http://www.sagemath.org/doc/reference/sage/graphs/graph_generators.html#sage.graphs.graph_generators.GraphGenerators.LCFGraph">sage</a>.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFCoxeterFruchtPowers1981" class="citation cs2"><a href="Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, Harold Scott MacDonald</a>; <a href="Robert_Frucht" title="Robert Frucht">Frucht, Roberto</a>; Powers, David L. (1981), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2BHjBQAAQBAJ&pg=PA13"><i>Zero-symmetric graphs</i></a>, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, p. 13, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-194580-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0658666">0658666</a></cite>.</span>

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<li id="cite_note-FOOTNOTECoxeterFruchtPowers1981Fig.&nbsp;1.1,_p.&nbsp;5-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeterFruchtPowers1981Fig.&nbsp;1.1,_p.&nbsp;5_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeterFruchtPowers1981">Coxeter, Frucht & Powers (1981)</a>, Fig. 1.1, p. 5.</span>

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<li id="cite_note-FOOTNOTECoxeterFruchtPowers198154-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeterFruchtPowers198154_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeterFruchtPowers1981">Coxeter, Frucht & Powers (1981)</a>, p. 54.</span>

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<li id="cite_note-FOOTNOTECoxeterFruchtPowers198112-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECoxeterFruchtPowers198112_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoxeterFruchtPowers1981">Coxeter, Frucht & Powers (1981)</a>, p. 12.</span>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-LCF_Notation"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LCFNotation.html">"LCF Notation"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li>

<li><cite id="CITEREFEd_Pegg_Jr.2003" class="citation cs2">Ed Pegg Jr. (29 December 2003), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130507063023/http://www.maa.org/editorial/mathgames/mathgames_12_29_03.html"><i>Math Games: Cubic Symmetric Graphs</i></a>, Mathematical Association of America, archived from <a rel="nofollow" class="external text" href="http://www.maa.org/editorial/mathgames/mathgames_12_29_03.html">the original</a> on 7 May 2013<span class="reference-accessdate">, retrieved <span class="nowrap">25 September</span> 2010</span></cite></li>

<li><a rel="nofollow" class="external text" href="http://bl.ocks.org/1703449">"Cubic Hamiltonian Graphs from LCF Notation"</a> – JavaScript interactive application, built with <a href="D3js" class="mw-redirect" title="D3js">D3js</a> library</li></ul>

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</style><div id="Graph_representations87" style="font-size:114%;margin:0 4em">Graph representations</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data structures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph (abstract data type)</a></li>

<li><a href="Adjacency_list" title="Adjacency list">Adjacency list</a></li>

<li><a href="Adjacency_matrix" title="Adjacency matrix">Adjacency matrix</a></li>

<li><a href="Incidence_matrix" title="Incidence matrix">Incidence matrix</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">XML-based formats</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li>DotML</li>

<li><a href="GraphML" title="GraphML">GraphML</a></li>

<li><a href="XGMML" title="XGMML">XGMML</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Text-based formats</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Graph_Modelling_Language" title="Graph Modelling Language">Graph Modelling Language</a> (GML)</li>

<li> for cubic Hamiltonian graphs</li>

<li><a href="Newick_format" title="Newick format">Newick format</a> for trees</li>

<li><a href="Trivial_Graph_Format" title="Trivial Graph Format">Trivial Graph Format</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Graph_database" title="Graph database">Graph database</a></li>

<li><a href="Graph_drawing" title="Graph drawing">Graph drawing</a></li>

<li><a href="Linked_data" title="Linked data">Linked data</a></li></ul>

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