The odd graph
of order
is a graph having vertices given by the -subsets of such that two vertices are connected by an edge
iff the associated subsets are disjoint (Biggs 1993, Ex.
8f, p. 58). Some care is needed since the convention of defining the odd graph
based on the -subsets
of is sometimes also used,
leading to a shifting of the index by one (e.g., West 2000, Ex. 1.1.28, p. 17).
By the definition of the odd graph using using the prevalent convention, the number of nodes in
is , where is a binomial coefficient.
For , 2, ..., the first few values are
1, 3, 10, 35, 126, ... (OEIS A001700).
is regular
of vertex degree and has graph diameter (Biggs 1976). The girth
of is 6 for (West 2000, p. 17; adjusting the indexing convention
to the more common definition based on subsets).
Balaban (1972) exhibited Hamiltonian cycles for and 5, Meredith and Lloyd (1972, 1973)
found cycles for
and 7, and Mather (1976) showed a Hamiltonian cycle
for (Shields and Savage).
Odd graphs are implemented in the Wolfram Language as FromEntity[Entity["Graph",
"Odd", n]],
and precomputed properties for small odd graphs are implemented as GraphData["Odd", n].
Balaban, A. T. "Chemical Graphs, Part XIII; Combinatorial Patterns." Rev. Roumaine Math. Pures Appl.17, 3-16, 1972.Biggs,
N. L. "Automorphic Graphs and the Krein Condition." Geom. Dedicata5,
117-127, 1976.Biggs, N. L. Algebraic
Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 161,
1993.Brouwer, A. E. "Odd Graphs." http://www.win.tue.nl/~aeb/drg/graphs/Odd.html.Brouwer,
A. E.; Cohen, A. M.; and Neumaier, A. Distance-Regular
Graphs. New York: Springer-Verlag, 1989.DistanceRegular.org.
"Odd Graphs." http://www.distanceregular.org/indexes/oddgraphs.html.Fiorini,
S. and Wilson, R. Edge-Colourings
of Graphs. Pittman, 1977.Mather, M. "The Rugby Footballers
of Croam." J. Combin. Theory Ser. B20, 62-63, 1976.Meredith,
G. H. J. and Lloyd, E. K. "The Hamiltonian graphs to ."
In Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972).
Southend: Inst. Math. Appl., pp. 229-236, 1972.Meredith, G. H. J.
and Lloyd, E. K. "The Footballers of Croam." J. Combin. Theory
Ser. B15, 161-166, 1973.Mütze, T. "On Hamilton
Cycles in Graphs Defined by Intersecting Set Systems." Not. Amer. Soc.74,
583-592, 2024.Shields, I. and Savage, C. D. "A Note on Hamilton
Cycles in Kneser Graphs." http://www.cybershields.com/publications/kneser3.pdf.Skiena,
S. Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, 1990.Sloane, N. J. A. Sequence A001700/M2848
in "The On-Line Encyclopedia of Integer Sequences."West, D. B.
"The Odd graph ."
Exercise 1.1.28 in Introduction
to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 17,
2000.