Odd graph

Definition and examples

The odd graph O_n has one vertex for each of the (n − 1)-element subsets of a (2n − 1)-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint. That is, O_n is a Kneser graph KG(2n − 1,n − 1).

O₂ is a triangle, while O₃ is the familiar Petersen graph.

The generalized odd graphs include the odd graphs and the folded cube graphs, and are defined as distance-regular graphs with diameter n − 1 and odd girth 2n − 1 for some n.

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 O₄. Odd graphs have been studied for their applications in chemical graph theory, in modeling the shifts of carbonium ions. They have also been proposed as a network topology in parallel computing.

The notation O_n for these graphs was introduced by Norman Biggs in 1972. Biggs and Tony Gardiner explain the name of odd graphs in an unpublished manuscript from 1974: each edge of an odd graph can be assigned the unique element of X which is the "odd man out", i.e., not a member of either subset associated with the vertices incident to that edge.

Properties

The odd graph O_n is regular of degree n. It has ( 2 n − 1 n − 1 ) vertices and n ( 2 n − 1 n − 1 ) / 2 edges. Therefore, the number of vertices for n = 1, 2,... is

Distance and symmetry

If two vertices in O_n correspond to sets that differ from each other by the removal of k elements from one set and the addition of k different elements, then they may be reached from each other in 2k steps, each pair of which performs a single addition and removal. If 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 O_n is n − 1.

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. Odd graphs are distance transitive, hence distance regular. 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. However, despite their high degree of symmetry, the odd graphs O_n for n > 2 are never Cayley graphs.

Odd graphs with n ≥ 3 have girth six; however, although they are not bipartite graphs, their odd cycles are much longer. Specifically, the odd graph O_n has odd girth 2n − 1. If a n-regular graph has diameter n − 1 and odd girth 2n − 1, and has only n distinct eigenvalues, it must be distance-regular. Distance-regular graphs with diameter n − 1 and odd girth 2n − 1 are known as the generalized odd graphs, and include the folded cube graphs as well as the odd graphs themselves.

Independent sets and vertex coloring

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

If I is a maximum independent set, formed by the sets that contain x, then the complement of I is the set of vertices that do not contain x. This complementary set induces a matching in G. Each vertex of the independent set is adjacent to n vertices of the matching, and each vertex of the matching is adjacent to n − 1 vertices of the independent set. 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 O_n is either n or n + 1, and in the case of the Petersen graph O₃ it is n + 1. When 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 n + 1. However, O₅, O₆, and O₇ can each be edge-colored with n colors.

Biggs 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 O₆, each weekday is represented by a color, and a 6-color edge coloring of O₆ provides a solution to the players' scheduling problem.

Hamiltonicity

The Petersen graph O₃ is a well known non-Hamiltonian graph, but O₄ through O₁₄ have been shown to contain Hamiltonian cycles. More strongly, combining the Hamiltonian cycle and edge coloring problems, it is possible to partition the edges of O_n (for n = 4, 5, 6, 7) into floor(n/2) Hamiltonian cycles; when n is odd, the leftover edges form a perfect matching. For n = 8, the odd number of vertices in O_n prevents an 8-color edge coloring from existing, but does not rule out the possibility of a partition into four Hamiltonian cycles.

The Lovász conjecture implies that every odd graph has a Hamiltonian path and that every odd graph O_n with n ≥ 4 has a Hamiltonian cycle.