On regular handicap graphs of even order

Regular handicap graphs of order $n \equiv 0$ (mod 8)

“…For any graph with a given regularity r and order n a simple counting argument shows the weight of each vertex i is already known as in the following lemma (see [8]).…”

“…A tournament in which this condition is satisfied, and every team plays r < n − 1 games is called a handicap incomplete round robin tournament. A summary of results of handicap tournaments obtained by the authors and other researchers can be found in [8]. In this paper we provide the details of the construction for n ≡ 0 (mod 8) for all feasible regularities.…”

A note on incomplete regular tournaments with handicap two of order n≡8(mod 16)

“…An overview of results on regular 1-handicap graphs of even order with additional references was recently accepted for publication [8]. For even-regular 1-handicap graphs of odd order, the results so far are sparse -see [4]. For d = 2, the author has studied handicap graphs with n ≡ 0 (mod 16) vertices [5,6].…”

“…Hence, a different method needs to be used. One can try methods similar to those used in papers referred to in [8].…”

Distance magic-type and distance antimagic-type labelings of graphs

“…[30] .For d = 1 and n odd, the existence of a 1-handicap tournament with at least one value of k is settled [28]. For d = 2, one class of n has been completely settled and one class partially settled (see Theorems 2.4.1, 2.4.2) [20, 21, 26].…”