Score sets in oriented 3-partite graphs
S. Pirzada, Merajuddin, T. A. Naikoo · Analysis in Theory and Applications · 2007
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 + and d − 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.