LCF notation

The Nauru graph[1] has LCF notation [5, −9, 7, −7, 9, −5]4.

In combinatorial mathematics, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian.[2][3] Since the graphs are Hamiltonian, 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. Often the pattern repeats, which is indicated by a superscript in the notation. For example, the Nauru graph,[1] shown on the right, has LCF notation [5, −9, 7, −7, 9, −5]4. Graphs may have different LCF notations, depending on precisely how the vertices are arranged.

The numbers between the square brackets are interpreted modulo N, where N is the number of vertices. Entries equal (modulo N) to 0, 1, and N−1 are not permitted,[4] since they do not correspond to valid third edges.

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

Name Vertices LCF notation
Tetrahedral graph 4 [2]4
Utility graph 6 [3]6
Cubical graph 8 [3,-3]4
Wagner graph 8 [4]8 or [4,-3,3,4]2
Bidiakis cube 12 [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 graph 12 [5,-5]6 or [-5,-3,3,5]3
Frucht graph 12 [-5,-2,-4,2,5,-2,2,5,-2,-5,4,2]
Truncated tetrahedral graph 12 [2,6,-2]4
Heawood graph 14 [5,-5]7
Möbius–Kantor graph 16 [5,-5]8
Pappus graph 18 [5,7,-7,7,-7,-5]3
Smallest zero-symmetric graph[7] 18 [5,-5]9
Desargues graph 20 [5,-5,9,-9]5
Dodecahedral graph 20 [10,7,4,-4,-7,10,-4,7,-7,4]2
McGee graph 24 [12,7,-7]8
Truncated cubical graph 24 [2,9,-2,2,-9,-2]4
Truncated octahedral graph 24 [3,-7,7,-3]6
Nauru graph 24 [5,-9,7,-7,9,-5]4
F26A graph 26 [-7, 7]13
Tutte–Coxeter graph 30 [-13,-9,7,-7,9,13]5
Dyck graph 32 [5,-5,13,-13]8
Gray graph 54 [-25,7,-7,13,-13,25]9
Truncated dodecahedral graph 60 [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 graph 70 [-29,-19,-13,13,21,-27,27,33,-13,13,19,-21,-33,29]5
Harries–Wong graph 70 [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-cage 70 [-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 graph 90 [17,-9,37,-37,9,-17]15
Biggs-Smith graph 102 [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-cage 112 [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 graph 112 [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-cage 126 [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. ↑ 1.0 1.1 Eppstein, D., The many faces of the Nauru graph on LiveJournal, 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.
  4. ↑ Klavdija Kutnar and Dragan Marušič, "Hamiltonicity of vertex-transitive graphs of order 4p," European Journal of Combinatorics, Volume 29, Issue 2 (February 2008), pp. 423-438, section 2.
  5. ↑ e.g. Maple, NetworkX, R, 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 658666.
  7. ↑ Coxeter, Frucht & Powers (1981), Fig. 1.1, p. 5.
  8. ↑ Coxeter, H. S. M.; Frucht, R.; Powers, D. L. (1981), Zero-symmetric graphs: trivalent graphical regular representations of groups, Academic Press, p. 54, ISBN 0-12-194580-4, MR 0658666
  9. ↑ Coxeter, Frucht & Powers (1981), p. 12.

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