LCF notation

In combinatorial mathematics, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle.[2][3] The cycle itself includes two out of the three adjacencies for each vertex, 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.

Description

In a Hamiltonian graph, the vertices can be arranged in a cycle, 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 modulo N, where N is the number of vertices. Entries congruent modulo N to 0, 1, or N  1 do not appear in this sequence of numbers,[4] because they would correspond either to a loop or multiple adjacency, neither of which are permitted in simple graphs.

Often the pattern repeats, and the number of repetitions can be indicated by a superscript in the notation. For example, the Nauru graph,[1] 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]4. A single graph may have multiple different LCF notations, depending on the choices of Hamiltonian cycle and starting vertex.

Applications

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.[5]

If a graph is represented by LCF notation, it is straightforward to test whether the graph is bipartite: this is true if and only if all of the offsets in the LCF notation are odd.[6]

Examples

NameVerticesLCF notation
Tetrahedral graph4[2]4
Utility graph6[3]6
Cubical graph8[3,−3]4
Wagner graph8[4]8 or [4,−3,3,4]2
Bidiakis cube12[6,4,−4]4 or [6,−3,3,6,3,−3]2 or [−3,6,4,−4,6,3,−4,6,−3,3,6,4]
Franklin graph12[5,−5]6 or [−5,−3,3,5]3
Frucht graph12[−5,−2,−4,2,5,−2,2,5,−2,−5,4,2]
Truncated tetrahedral graph12[2,6,−2]4
Heawood graph14[5,−5]7
Möbius–Kantor graph16[5,−5]8
Pappus graph18[5,7,−7,7,−7,−5]3
Smallest zero-symmetric graph[7]18[5,−5]9
Desargues graph20[5,−5,9,−9]5
Dodecahedral graph20[10,7,4,−4,−7,10,−4,7,−7,4]2
McGee graph24[12,7,−7]8
Truncated cubical graph24[2,9,−2,2,−9,−2]4
Truncated octahedral graph24[3,−7,7,−3]6
Nauru graph24[5,−9,7,−7,9,−5]4
F26A graph26[−7, 7]13
Tutte–Coxeter graph30[−13,−9,7,−7,9,13]5
Dyck graph32[5,−5,13,−13]8
Gray graph54[−25,7,−7,13,−13,25]9
Truncated dodecahedral graph60[30, −2, 2, 21, −2, 2, 12, −2, 2, −12, −2, 2, −21, −2, 2, 30, −2, 2, −12, −2, 2, 21, −2, 2, −21, −2, 2, 12, −2, 2]2
Harries graph70[−29,−19,−13,13,21,−27,27,33,−13,13,19,−21,−33,29]5
Harries–Wong graph70[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]
Balaban 10-cage70[−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]
Foster graph90[17,−9,37,−37,9,−17]15
Biggs–Smith graph102[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]
Balaban 11-cage112[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]
Ljubljana graph112[47, −23, −31, 39, 25, −21, −31, −41, 25, 15, 29, −41, −19, 15, −49, 33, 39, −35, −21, 17, −33, 49, 41, 31, −15, −29, 41, 31, −15, −25, 21, 31, −51, −25, 23, 9, −17, 51, 35, −29, 21, −51, −39, 33, −9, −51, 51, −47, −33, 19, 51, −21, 29, 21, −31, −39]2
Tutte 12-cage126[17, 27, −13, −59, −35, 35, −11, 13, −53, 53, −27, 21, 57, 11, −21, −57, 59, −17]7

Extended LCF notation

A more complex extended version of LCF notation was provided by Coxeter, Frucht, and Powers in later work.[8] 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]4, and so can be written [5, 9, 7; ]4 in the extended notation.[9]

References

  1. Eppstein, D., The many faces of the Nauru graph, 2007.
  2. Pisanski, Tomaž; Servatius, Brigitte (2013), "2.3.2 Cubic graphs and LCF notation", Configurations from a Graphical Viewpoint, Springer, p. 32, ISBN 9780817683641.
  3. Frucht, R. (1976), "A canonical representation of trivalent Hamiltonian graphs", Journal of Graph Theory, 1 (1): 45–60, doi:10.1002/jgt.3190010111, MR 0463029.
  4. Kutnar, Klavdija; Marušič, Dragan (2008), "Hamiltonicity of vertex-transitive graphs of order 4p", European Journal of Combinatorics, 29 (2): 423–438, arXiv:math/0606585, doi:10.1016/j.ejc.2007.02.002, MR 2388379. See Section 2.
  5. e.g. Maple, NetworkX, igraph, and sage.
  6. Coxeter, Harold Scott MacDonald; Frucht, Roberto; Powers, David L. (1981), Zero-symmetric graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, p. 13, ISBN 0-12-194580-4, MR 0658666.
  7. Coxeter, Frucht & Powers (1981), Fig. 1.1, p. 5.
  8. Coxeter, Frucht & Powers (1981), p. 54.
  9. Coxeter, Frucht & Powers (1981), p. 12.
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