Score Sets in Oriented Graphs

and d−v are the outdegree and indegree respectively of v and n is the number of vertices in D. The set of distinct scores of the vertices in an oriented graph D is called its score set. If a > 0 and d > 1 are positive integers, we show there exists an oriented graph with score set {a, ad, ad, . . . , ad} except for a = 1, d = 2, n > 0, and for a = 1, d = 3, n > 0. It is also shown that there exists no oriented graph with score set {a, ad, ad, . . . , ad}, n > 0 when either a = 1, d = 2, or a = 1, d = 3. Also we prove for the non-negative integers a1, a2, . . . , an with a1 < a2 < · · · < an, there is always an oriented graph with an + 1 vertices with score set {a ′ 1, a ′ 2, . . . , a ′ n}, where