On regular d-handicap tournaments - Bryan Freyberg; Melissa Keranen (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2023)
A k-regular d-handicap tournament is an incomplete tournament in which n teams, ranked according to the natural numbers, play exactly k < n − 1 different teams exactly once and the strength of schedule of the ith ranked team is d more than the (i − 1)st ranked team for some d ≥ 1. That is, strength of schedules increase arithmetically by d with strength of team. A d-handicap distance antimagic labeling of a graph G = (V,E) of order n is a bijection ℓ : V → 1,2,…,n with induced weight function w(xi)=Σ xj ∈ N(xi)l(xj) such that ℓ(xi)=i and the sequence of weights w(x1),w(x2),…,w(xn) forms an arithmetic sequence with difference d ≥ 1. A graph G which admits such a labeling is called a d-handicap graph.Constructing a k-regular d-handicap tournament on n teams is equivalent to finding a k-regular d-handicap graph of order n. For d = 1 and n even, the existence has recently been completely settled for all pairs (n,k), and some results are known for d = 2. For d > 2, the only known result is restricted to the case where n is divisible by 2d + 2. In this paper, we construct infinite families of d-handicap graphs where the order is not restricted to a power of 2.