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 dx^＋ and dui^- be the outdegree and indegree ofx respectively. Define aui （o...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 dx^＋ and dui^- be the outdegree and indegree ofx respectively. Define aui （or simply ai） = q ＋ r ＋ dui^＋ - dui^-, bvj （or simply b j） = p ＋ r ＋ d^＋vj - d^-vj and cwk （or simply ck） =p ＋ q ＋ dwk^＋ -dwk^- as the scores of ui in U,vj in V and wk 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 a1 is a non-negative integer, ai（2 ≤ i ≤ n - 1） are even positive integers and an is any positive integer, then for n 〉 3, there exists an oriented 3-partite graph with the score set A ={a1,Σ2i=1 ai,…,Σni=1 ai}, except when A = {0, 2, 3}. Some more results for score sets in oriented 3-partite graphs are obtained.展开更多
文摘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 dx^＋ and dui^- be the outdegree and indegree ofx respectively. Define aui （or simply ai） = q ＋ r ＋ dui^＋ - dui^-, bvj （or simply b j） = p ＋ r ＋ d^＋vj - d^-vj and cwk （or simply ck） =p ＋ q ＋ dwk^＋ -dwk^- as the scores of ui in U,vj in V and wk 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 a1 is a non-negative integer, ai（2 ≤ i ≤ n - 1） are even positive integers and an is any positive integer, then for n 〉 3, there exists an oriented 3-partite graph with the score set A ={a1,Σ2i=1 ai,…,Σni=1 ai}, except when A = {0, 2, 3}. Some more results for score sets in oriented 3-partite graphs are obtained.
