Odd graph

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]

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.

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.

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.