TY - JOUR

T1 - On the existence of semigraphs and complete semigraphs with given parameters

AU - Shetty, Jyoti

AU - Sudhakara, G.

AU - Madhusudanan, Vinay

N1 - Publisher Copyright:
© 2021 THE AUTHORS

PY - 2021/12

Y1 - 2021/12

N2 - E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete. Let P be a set with p elements and {E1,E2,…,Eq} be a collection of subsets of P with ⋃i=1qEi=P. We derive an expression for the maximum value of the difference ∑j=1k|Eij|-⋃i=1kEij for 2⩽k⩽q, where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges {e1,e2,…,eq}, where the set Ei is the set of vertices on the edge ei,1⩽i⩽q. Combining the above two results, we characterize a complete semigraph.

AB - E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete. Let P be a set with p elements and {E1,E2,…,Eq} be a collection of subsets of P with ⋃i=1qEi=P. We derive an expression for the maximum value of the difference ∑j=1k|Eij|-⋃i=1kEij for 2⩽k⩽q, where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges {e1,e2,…,eq}, where the set Ei is the set of vertices on the edge ei,1⩽i⩽q. Combining the above two results, we characterize a complete semigraph.

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U2 - 10.1016/j.asej.2021.04.002

DO - 10.1016/j.asej.2021.04.002

M3 - Article

AN - SCOPUS:85106874178

SN - 2090-4479

JO - Ain Shams Engineering Journal

JF - Ain Shams Engineering Journal

ER -