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Odd Graph


The odd graph O_n of order n is a graph having vertices given by the (n-1)-subsets of {1,...,2n-1} 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 n-subsets of {1,...,2n+1} 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 O_n is (2n-1; n-1), where (n; k) is a binomial coefficient. For n=1, 2, ..., the first few values are 1, 3, 10, 35, 126, ... (OEIS A001700).

OddGraphs

O_1 is isomorphic to the singleton graph, O_2 to the triangle graph C_3, and O(3) to the Petersen graph (Skiena 1990, p. 162). The Kneser graph K(n,k) is a generalization of the odd graph, with O_n corresponding to K(2n-1,n-1). The bipartite Kneser graph is a generalization of the bipartite double graph of the odd graph, with O_n corresponding to H(2n-1,n-1) (which, like O_n, is distance-transitive; Brouwer et al. 1989, p. 222).

O_n is regular of vertex degree n and has graph diameter n-1 (Biggs 1976). The girth of O_n is 6 for n>=4 (West 2000, p. 17; adjusting the indexing convention to the more common definition based on n-1 subsets).

The odd graphs are distance-transitive, and therefore also distance-regular. They are also automorphic graphs (Biggs 1976). It is conjectured that O_n is of class 1 except for the cases n=3 and n a power of two (Fiorini and Wilson 1977).

Balaban (1972) exhibited Hamiltonian cycles for n=4 and 5, Meredith and Lloyd (1972, 1973) found cycles for n=6 and 7, and Mather (1976) showed a Hamiltonian cycle for n=8 (Shields and Savage).

Since the odd graph is a special case of the Kneser graph, its independence number follows from the value for alpha(K(n,k)) as

 alpha(O_n)=(2n-2; n-2).

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}].


See also

Complete Graph, Distance-Regular Graph, Distance-Transitive Graph, Kneser Graph, Odd Vertex, Petersen Graph

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References

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. Dedicata 5, 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. B 20, 62-63, 1976.Meredith, G. H. J. and Lloyd, E. K. "The Hamiltonian graphs O_4 to O_7." 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. B 15, 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 O_k." Exercise 1.1.28 in Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, p. 17, 2000.

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Odd Graph

Cite this as:

Weisstein, Eric W. "Odd Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OddGraph.html

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