Abstract
Let D(U,V,W) be an oriented 3-partite graph with |U| = p, |V| = q and |W| = r. For any vertex x in D(U,V,W), let d + x and d − x be the outdegree and indegree of x respectively. Define \(a_{u_i } \) (or simply a i ) = q + r + \(d_{u_i }^ + - d_{u_i }^ - \), \(b_{v_j } \) (or simply b j ) = p + r + d + ν j − \(d_{v_j }^ - \) and \(c_{w_k } \) (or simply c k ) = p + q + \(d_{w_k }^ + - d_{w_k }^ - \) as the scores of u i in U,v j in V and w k in W respectively. The set A of distinct scores of the vertices of D(U,V,W) is called its score set. In this paper, we prove that if a 1 is a non-negative integer, a i (2 ≤ i ≤n − 1) are even positive integers and a n is any positive integer, then for n ≥ 3, there exists an oriented 3-partite graph with the score set \(A = \left\{ {a_1 ,\sum\limits_{i = 1}^2 {a_i , \cdots ,} \sum\limits_{i = 1}^n {a_i } } \right\}\), except when A = {0,2,3}. Some more results for score sets in oriented 3-partite graphs are obtained.
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Pirzada, S., Merajuddin & Naikoo, T.A. Score sets in oriented 3-partite graphs. Anal. Theory Appl. 23, 363–374 (2007). https://doi.org/10.1007/s10496-007-0363-7
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DOI: https://doi.org/10.1007/s10496-007-0363-7