Odd graph

Odd graph
${\displaystyle O_{3}=KG(5,2)}$ is the Petersen graph
Vertices	${\displaystyle {\tbinom {2n-1}{n-1}}}$
Edges	${\displaystyle n{\tbinom {2n-1}{n-1}}/2}$
Diameter	${\displaystyle n-1}$ [1] [2]
Girth	3 for ${\displaystyle O_{2}}$ 5 for ${\displaystyle O_{3}}$ 6 otherwise [3]
Properties	Distance-transitive
Notation	On
Table of graphs and parameters

The odd graph ${\displaystyle O_{4}=KG(7,3)}$

In the mathematical field of graph theory, the odd graphs are a family of symmetric graphs with high odd girth, defined from certain set systems. They include and generalize the Petersen graph.

Definition and examples

The odd graph ${\displaystyle O_{n}}$ has one vertex for each of the ${\displaystyle (n-1)}$-element subsets of a ${\displaystyle (2n-1)}$-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint. ^[2] That is, ${\displaystyle O_{n}}$ is the Kneser graph ${\displaystyle KG(2n-1,n-1)}$.

${\displaystyle O_{2}}$ is a triangle, while ${\displaystyle O_{3}}$ is the familiar Petersen graph.

The generalized odd graphs are defined as distance-regular graphs with diameter ${\displaystyle n-1}$ and odd girth ${\displaystyle 2n-1}$ for some ${\displaystyle n}$. ^[4] They include the odd graphs and the folded cube graphs.

History and applications

Although the Petersen graph has been known since 1898, its definition as an odd graph dates to the work of Kowalewski (1917), who also studied the odd graph ${\displaystyle O_{4}}$. ^{[2] [5]} Odd graphs have been studied for their applications in chemical graph theory, in modeling the shifts of carbonium ions. ^{[6] [7]} They have also been proposed as a network topology in parallel computing. ^[8]

The notation ${\displaystyle O_{n}}$ for these graphs was introduced by Norman Biggs in 1972. ^[9] Biggs and Tony Gardiner explain the name of odd graphs in an unpublished manuscript from 1974: each edge of an odd graph can be assigned the unique element which is the "odd man out", i.e., not a member of either subset associated with the vertices incident to that edge. ^{[10] [11]}

Properties

The odd graph ${\displaystyle O_{n}}$ is regular of degree ${\displaystyle n}$. It has ${\displaystyle {\tbinom {2n-1}{n-1}}}$ vertices and ${\displaystyle n{\tbinom {2n-1}{n-1}}/2}$ edges. Therefore, the number of vertices for ${\displaystyle n=1,2,\dots }$ is

1, 3, 10, 35, 126, 462, 1716, 6435 (sequence A001700 in the OEIS).

Distance and symmetry

If two vertices in ${\displaystyle O_{n}}$ correspond to sets that differ from each other by the removal of ${\displaystyle k}$ elements from one set and the addition of ${\displaystyle k}$ different elements, then they may be reached from each other in ${\displaystyle 2k}$ steps, each pair of which performs a single addition and removal. If ${\displaystyle 2k<n}$, this is a shortest path; otherwise, it is shorter to find a path of this type from the first set to a set complementary to the second, and then reach the second set in one more step. Therefore, the diameter of ${\displaystyle O_{n}}$ is ${\displaystyle n-1}$. ^{[1] [2]}

Every odd graph is 3-arc-transitive: every directed three-edge path in an odd graph can be transformed into every other such path by a symmetry of the graph. ^[12] Odd graphs are distance transitive, hence distance regular. ^[2] As distance-regular graphs, they are uniquely defined by their intersection array: no other distance-regular graphs can have the same parameters as an odd graph. ^[13] However, despite their high degree of symmetry, the odd graphs ${\displaystyle O_{n}}$ for ${\displaystyle n>2}$ are never Cayley graphs. ^[14]

Because odd graphs are regular and edge-transitive, their vertex connectivity equals their degree, ${\displaystyle n}$. ^[15]

Odd graphs with ${\displaystyle n>3}$ have girth six; however, although they are not bipartite graphs, their odd cycles are much longer. Specifically, the odd graph ${\displaystyle O_{n}}$ has odd girth ${\displaystyle 2n-1}$. If an ${\displaystyle n}$-regular graph has diameter ${\displaystyle n-1}$ and odd girth ${\displaystyle 2n-1}$, and has only ${\displaystyle n}$ distinct eigenvalues, it must be distance-regular. Distance-regular graphs with diameter ${\displaystyle n-1}$ and odd girth ${\displaystyle 2n-1}$ are known as the generalized odd graphs, and include the folded cube graphs as well as the odd graphs themselves. ^[4]

Independent sets and vertex coloring

Let ${\displaystyle O_{n}}$ be an odd graph defined from the subsets of a ${\displaystyle (2n-1)}$-element set ${\displaystyle X}$, and let ${\displaystyle x}$ be any member of ${\displaystyle X}$. Then, among the vertices of ${\displaystyle O_{n}}$, exactly ${\displaystyle {\tbinom {2n-2}{n-2}}}$ vertices correspond to sets that contain ${\displaystyle x}$. Because all these sets contain ${\displaystyle x}$, they are not disjoint, and form an independent set of ${\displaystyle O_{n}}$. That is, ${\displaystyle O_{n}}$ has ${\displaystyle 2n-1}$ different independent sets of size ${\displaystyle {\tbinom {2n-2}{n-2}}}$. It follows from the Erdős–Ko–Rado theorem that these are the maximum independent sets of ${\displaystyle O_{n}}$. that is, the independence number of ${\displaystyle O_{n}}$ is ${\displaystyle {\tbinom {2n-2}{n-2}}.}$ Further, every maximum independent set must have this form, so ${\displaystyle O_{n}}$ has exactly ${\displaystyle 2n-1}$ maximum independent sets. ^[2]

If ${\displaystyle I}$ is a maximum independent set, formed by the sets that contain ${\displaystyle x}$, then the complement of ${\displaystyle I}$ is the set of vertices that do not contain ${\displaystyle x}$. This complementary set induces a matching in ${\displaystyle G}$. Each vertex of the independent set is adjacent to ${\displaystyle n}$ vertices of the matching, and each vertex of the matching is adjacent to ${\displaystyle n-1}$ vertices of the independent set. ^[2] Because of this decomposition, and because odd graphs are not bipartite, they have chromatic number three: the vertices of the maximum independent set can be assigned a single color, and two more colors suffice to color the complementary matching.

Edge coloring

By Vizing's theorem, the number of colors needed to color the edges of the odd graph ${\displaystyle O_{n}}$ is either ${\displaystyle n}$ or ${\displaystyle n+1}$, and in the case of the Petersen graph ${\displaystyle O_{3}}$ it is ${\displaystyle n+1}$. When ${\displaystyle n}$ is a power of two, the number of vertices in the graph is odd, from which it again follows that the number of edge colors is ${\displaystyle n+1}$. ^[16] However, ${\displaystyle O_{5}}$, ${\displaystyle O_{6}}$, and ${\displaystyle O_{7}}$ can each be edge-colored with ${\displaystyle n}$ colors. ^{[2] [16]}

Biggs ^[9] explains this problem with the following story: eleven soccer players in the fictional town of Croam wish to form up pairs of five-man teams (with an odd man out to serve as referee) in all 1386 possible ways, and they wish to schedule the games between each pair in such a way that the six games for each team are played on six different days of the week, with Sundays off for all teams. Is it possible to do so? In this story, each game represents an edge of ${\displaystyle O_{6}}$, each weekday is represented by a color, and a 6-color edge coloring of ${\displaystyle O_{6}}$ provides a solution to the players' scheduling problem.

Hamiltonicity

The Petersen graph ${\displaystyle O_{3}}$ is a well known non-Hamiltonian graph, but all odd graphs ${\displaystyle O_{n}}$ for ${\displaystyle n\geq 4}$ are known to have a Hamiltonian cycle. ^[17] As the odd graphs are vertex-transitive, they are thus one of the special cases with a known positive answer to Lovász' conjecture on Hamiltonian cycles in vertex-transitive graphs. Biggs ^[2] conjectured more generally that the edges of ${\displaystyle O_{n}}$ can be partitioned into ${\displaystyle \lfloor n/2\rfloor }$ edge-disjoint Hamiltonian cycles. When ${\displaystyle n}$ is odd, the leftover edges must then form a perfect matching. This stronger conjecture was verified for ${\displaystyle n=4,5,6,7}$. ^{[2] [16]} For ${\displaystyle n=8}$, the odd number of vertices in ${\displaystyle O_{n}}$ prevents an 8-color edge coloring from existing, but does not rule out the possibility of a partition into four Hamiltonian cycles.

References

1. Biggs, Norman L. (1976), "Automorphic graphs and the Krein condition", Geometriae Dedicata, 5 (1): 117–127, doi:10.1007/BF00148146 .
2. Biggs, Norman (1979), "Some odd graph theory", Second International Conference on Combinatorial Mathematics, Annals of the New York Academy of Sciences, 319: 71–81, doi:10.1111/j.1749-6632.1979.tb32775.x .
3. West, Douglas B. (2000), "Exercise 1.1.28", Introduction to Graph Theory (2nd ed.), Englewood Cliffs, NJ: Prentice-Hall, p. 17.
4. Van Dam, Edwin; Haemers, Willem H. (2010), An Odd Characterization of the Generalized Odd Graphs, CentER Discussion Paper Series No. 2010-47, SSRN   1596575 .
5. Kowalewski, A. (1917), "W. R. Hamilton's Dodekaederaufgabe als Buntordnungproblem", Sitzungsber. Akad. Wiss. Wien (Abt. IIa), 126: 67–90, 963–1007. As cited by Biggs (1979).
6. Balaban, Alexandru T.; Fǎrcaşiu, D.; Bǎnicǎ, R. (1966), "Graphs of multiple 1, 2-shifts in carbonium ions and related systems", Rev. Roum. Chim., 11: 1205.
7. Balaban, Alexandru T. (1972), "Chemical graphs, Part XIII: Combinatorial patterns", Rev. Roumaine Math. Pures Appl., 17: 3–16.
8. Ghafoor, Arif; Bashkow, Theodore R. (1991), "A study of odd graphs as fault-tolerant interconnection networks", IEEE Transactions on Computers, 40 (2): 225–232, doi:10.1109/12.73594 .
9. Biggs, Norman (1972), Guy, Richard K. (ed.), "An edge-colouring problem", Research Problems, American Mathematical Monthly, 79 (9): 1018–1020, doi:10.2307/2318076, JSTOR   2318076 .
10. Brouwer, Andries; Cohen, Arjeh M.; Neumaier, A. (1989), Distance-regular Graphs, ISBN   0-387-50619-5 .
11. Ed Pegg, Jr. (December 29, 2003), Cubic Symmetric Graphs, Math Games, Mathematical Association of America, archived from the original on August 21, 2010, retrieved August 24, 2010.
12. Babai, László (1995), "Automorphism groups, isomorphism, reconstruction", in Graham, Ronald L.; Grötschel, Martin; Lovász, László (eds.), Handbook of Combinatorics, vol. I, North-Holland, pp. 1447–1540, Proposition 1.9, archived from the original on June 11, 2010.
13. Moon, Aeryung (1982), "Characterization of the odd graphs O_k by parameters", Discrete Mathematics, 42 (1): 91–97, doi: 10.1016/0012-365X(82)90057-7 .
14. Godsil, C. D. (1980), "More odd graph theory", Discrete Mathematics, 32 (2): 205–207, doi: 10.1016/0012-365X(80)90055-2 .
15. Watkins, Mark E. (1970), "Connectivity of transitive graphs", Journal of Combinatorial Theory , 8: 23–29, doi: 10.1016/S0021-9800(70)80005-9 , MR   0266804
16. Meredith, Guy H. J.; Lloyd, E. Keith (1973), "The footballers of Croam", Journal of Combinatorial Theory, Series B, 15 (2): 161–166, doi: 10.1016/0095-8956(73)90016-6 .
17. Mütze, Torsten; Nummenpalo, Jerri; Walczak, Bartosz (2018), "Sparse Kneser graphs are Hamiltonian", STOC'18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, New York: ACM, pp. 912–919, arXiv: 1711.01636 , MR   3826304

External links

• Weisstein, Eric W., "Odd Graph", MathWorld