Score Sets in Oriented k - partite Graphs
نویسندگان
چکیده
Let D(U1, U2, ..., Uk) be an oriented k -partite graph with |Ui| = ni, 1 ≤ i ≤ k. Then, the score of a vertex ui in Ui is defined by aui (or simply ai ) = ∑k t=1, t 6=i nt+d + ui −d − ui , where d+ui and d − ui are respectively the outdegree and indegree of ui. The set A of distinct scores of the vertices of D(U1, U2, ..., Uk) is called its score set. In this paper, we prove that if a1 is a non-negative integer, ai (2 ≤ i ≤ n− 1) are even positive integers and an is any positive integer, then for every n ≥ k ≥ 3, there exists an oriented k -partite graph with score set A = { a1, ∑2 i=1 ai, ..., ∑n i=1 ai } , except when a1 = 0, ak = 1, n = k ≥ 3.
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