One-factorizations of the complete graph Kp+1 arising from parabolas

There are three types of affine regular polygons in AG(2, q): ellipse, hyperbola and parabola. The first two cases have been investigated in previous papers. In this note, a particular class of geometric one-factorizations of the complete graph Kn arising from parabolas is constructed and described in full detail. With the support of computer aided investigation, it is also conjectured that up to isomorphisms this is the only one-factorization where each one-factor is either represented by a line or a parabola.


Introduction
For a positive even integer n, a one-factorization of the complete graph K n is a partition of the edge set into n − 1 one-factors-each consisting of n 2 edges partitioning the vertex set. One-factorizations of complete graphs play a crucial role in many practical applications, like for instance scheduling tournaments, where a round robin tournament is to be played in the minimum number of sessions. Besides applications, one-factorizations have strong connections to Design Theory; see for instance [13].
Our approach to the problem of constructing one-factorizations of complete graphs is essentially geometric, as in [3,6,9,10], and is based on techniques that have previously been used to find one-factorizations of multigraphs; see for instance [2,4,7,11].
Basically, there are three types of affine regular polygons in the finite affine plane AG(2, q). One-factorizations arising from ellipses and hyperbolas have already been addressed in [6,9]. In this paper the remaining case, the parabola, is investigated.
Our main result is the construction of a parabolic one-factorization-that is, a one-factorization where all one-factors except one are represented by parabolas, and the remaining one is represented by a line-for every complete graph K p+1 with p an odd prime. We may also provide a classification of parabolic one-factorizations.
Our notation is standard. For general information about one-factorizations of complete graphs see for instance [8,12,13].

Preliminaries
Henceforth we assume that p ≥ 3 is a prime number. We fix a projective frame in PG(2, p) with homogeneous coordinates (X 0 :X 1 :X 2 ), and consider PG(2, p) as AG(2, p) ∪ ℓ ∞ where ℓ ∞ has equation X 0 = 0. As usual, the points of AG(2, p) are written as (X, Y ) with X = X 1 X 0 and Y = X 2 X 0 . In AG(2, p), let P a be the parabola with affine equation Y = X 2 + a, where a varies in Z p , and V ∞ = (0:0:1) the point at infinity of the line X 1 = 0. Note that, in the projective closure of AG(2, p), any two parabolas P a and P b , with a $ = b, meet at the point V ∞ only.
Let V i = (i, i 2 ) denote the points on P 0 for i = 0, 1, . . . , p − 1. For k = 1, 2, . . . , p−1 2 , let P k i denote the pole of the line V i V i+k with respect to P 0 . The equation of the tangent line t i to P 0 at V i is see Figure 1. Further, let P ∞ i denote the point at infinity of the line t i , that is, P ∞ i = (0:1:2i). Lemma 2.1. For a fixed k, the points P k 0 , P k 1 , . . . , P k p−1 are on the parabola P Proof. The claim follows from the equality The vertices of the complete graph K p+1 correspond to the points of P 0 ∪ {V ∞ }, while the edges of K p+1 correspond to the points of type P k i , with k = 1, 2, . . . , p−1 2 , ∞. Thus the set of edges of K p+1 corresponds to the set of points These points are called external points with respect to P 0 . In this setting, a one-factor of K p+1 is a set consisting of p+1 2 points of type P k i , for i ∈ {0, 1, . . . , p − 1} and k ∈ * 1, 2, . . . , p−1 2 + ∪ {∞}, satisfying the tangent property, that is, no tangent to P 0 meets the set in more than one point; see [6]. Then, a one-factorization of K p+1 is just a partition of all the points of type P k i into p one-factors.

Results
Remark that a parabola of type P a cannot contain any point of type P ∞ j , therefore a subset of its points satisfying the tangent property consists of at most p−1 2 points. If the line ℓ is not a tangent to P 0 , then ℓ is called a secant if |ℓ ∩ P 0 | = 2 and ℓ is called an external line if |ℓ ∩ P 0 | = 0. It is well known (see e.g. [5, Lemma 6.14]) that a secant contains p−1 2 points of E and an external line contains p+1 2 points of E. These motivate the following definitions.
points of type P k j on P a , together with a suitable point at infinity. A onefactor so defined is referred to as a parabolic one-factor.  Proof. The proof is constructive. Let The set F 0 is a one-factor represented by the secant line of P 0 of equation X = 0, and P ∞ 0 is its pole with respect to P 0 .
For k = 1, 2, . . . , p−1 2 , define the following sets of points: are disjoint subsets of the parabola P in two points, P k i and P k i+k . One of these points falls in G k , the other one in H k , and the claim follows.
Parabolic one-factorisations are completely characterised in the projective closure of AG(2, p).
Theorem 3.5. Let p > 5 be an odd prime and F be a parabolic onefactorization of the complete graph K p+1 . Then F is isomorphic to the onefactorization constructed in Theorem 3.4.
Proof. Let ℓ be the line representing the unique linear one-factor of F and L denote the pole of ℓ with respect to P 0 . First, we show that ℓ contains the point V ∞ . By definition, ℓ ∪ {L} must contain one affine point from each parabola of type P a . Hence ℓ must be a tangent to at least p−1 2 − 1 > 1 parabolas of type P a . Suppose that the affine equation of ℓ is Y = mX + b. Then ℓ contains exactly one point of P a if and only if the discriminant of the quadratic equation X 2 − mX + a − b = 0 is zero, that is, (1) From (1), the line ℓ would be a tangent to at most one parabola of type P a , hence it must be assumed that the affine equation of ℓ is of type X = c. Now consider the linear transformation ϕ ∈ PGL(3, p) associated to the matrix  Then (1 : c : c 2 ) ϕ = (1 : 0 : 0) and (0 : 0 : 1) ϕ = (0 : 0 : 1). Hence, the unique linear one-factor of F ϕ corresponds, by projectivity, to the line X = 0, that is, the set of points Further, the linear transformation ϕ fixes every parabola P a setwise since (1 : For a fixed k ∈ {1, 2, . . . , p−1 2 } let G k and H k denote the two one-factors of F ϕ which are represented by the parabola P We may assume without loss of generality that it belongs to G k . Then, by the tangent property, P k k 2 +k must belong to H k . For j = 1, . . . , p−3 2 , the points P k k 2 +2jk must belong to G k , while the points P k is in H k . Thus, F ϕ is the one-factorization constructed in Theorem 3.4 and hence F is isomorphic to F ϕ .
We conclude with a conjecture that is supported by our computer aided investigations. With the aid of Magma [1] we verified that the conjecture holds true for p ≤ 17. Conjecture 3.6. Let p > 7 be an odd prime, F be a one-factorization of the complete graph K p+1 such that each one-factor of F is either represented by a line or a parabola. Then F is either a parabolic one-factorization or each one-factor of F is represented by a line.

Examples for small p
The examples described in this section serve to illustrate the results from the previous sections.

p = 7
Let us consider the parabola P 0 of projective equation X 0 X 2 = X 2 1 in PG (2,7). The construction in Theorem 3.4 provides the following partition of the points of type P k i : • F 0 is represented by the secant line X 1 = 0,