Score sets in oriented 3-partite graphs |
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| Authors: | S Pirzada Merajuddin T A Naikoo |
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| Institution: | (1) Department of Mathematics, University of Kashmir, Srinagar, 190006, India;(2) Department of Applied Mathematics, F/O Engineering and Tech., A.M.U. Aligarh, India;(3) Department of Mathematics, University of Kashmir, Srinagar, 190006, India |
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| 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
(or simply a
i
) = q + r +
,
(or simply b
j
) = p + r + d
+
ν
j
−
and
(or simply c
k
) = p + q +
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
, except when A = {0,2,3}. Some more results for score sets in oriented 3-partite graphs are obtained.
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| Keywords: | oriented graph oriented 3-partite graph tournament score set GRAPHS ORIENTED SETS results sets graphs integers positive prove paper score vertex oriented |
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