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Motzkin and Straus established a close connection between the maximum clique problem and a solution (namely graph-Lagrangian) to the maximum value of a class of homogeneous quadratic multilinear functions over the standard simplex of the Euclidean space in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique problem in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we develop a homogeneous multilinear function based on the structure of hypergraphs and their complement hypergraphs. Its maximum value generalizes the graph-Lagrangian. Specifically, we establish a connection between the clique number and the generalized graph-Lagrangian of $3$-uniform graphs, which supports the conjecture posed in this paper.
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참고문헌 (24)
참고문헌 신청
A. F. Sidorenko / 1987 / On the maximal number of edges in a homogeneous hypergraph that does not contain prohibited subgraphs / Mat. Zametki 41 (3) : 433 ~ 455
D. Mubayi / 2006 / A hypergraph extension of Tur´an’s theorem / J. Combin. Theory Ser. B 96 (1) : 122 ~ 134
I. M. Bomze / 1997 / Evolution towards the maximum clique / J. Global Optim 10 (2) : 143 ~ 164
L. E. Gibbons / 1997 / Continuous characterizations of the maximum clique problem / Math. Oper. Res 22 (3) : 754 ~ 768
M. Budinich / 2003 / Exact bounds on the order of the maximum clique of a graph / Discrete Appl. Math 127 (3) : 535 ~ 543
D. Mubayi / 2006 / A hypergraph extension of Tur´an’s theorem / J. Combin. Theory Ser. B 96 (1) : 122 ~ 134
I. M. Bomze / 1997 / Evolution towards the maximum clique / J. Global Optim 10 (2) : 143 ~ 164
L. E. Gibbons / 1997 / Continuous characterizations of the maximum clique problem / Math. Oper. Res 22 (3) : 754 ~ 768
M. Budinich / 2003 / Exact bounds on the order of the maximum clique of a graph / Discrete Appl. Math 127 (3) : 535 ~ 543
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