TY - JOUR
T1 - Orthonormal Representations of H-Free Graphs
AU - Balla, Igor
AU - Letzter, Shoham
AU - Sudakov, Benny
N1 - Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/10/1
Y1 - 2020/10/1
N2 - Let x1, … , xn∈ Rd be unit vectors such that among any three there is an orthogonal pair. How large can n be as a function of d, and how large can the length of x1+ ⋯ + xn be? The answers to these two celebrated questions, asked by Erdős and Lovász, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovász ϑ-function and minimum semidefinite rank. In this paper, we study these parameters for general H-free graphs. In particular, we show that for certain bipartite graphs H, there is a connection between the Turán number of H and the maximum of ϑ(G¯) over all H-free graphs G.
AB - Let x1, … , xn∈ Rd be unit vectors such that among any three there is an orthogonal pair. How large can n be as a function of d, and how large can the length of x1+ ⋯ + xn be? The answers to these two celebrated questions, asked by Erdős and Lovász, are closely related to orthonormal representations of triangle-free graphs, in particular to their Lovász ϑ-function and minimum semidefinite rank. In this paper, we study these parameters for general H-free graphs. In particular, we show that for certain bipartite graphs H, there is a connection between the Turán number of H and the maximum of ϑ(G¯) over all H-free graphs G.
KW - Lovász ϑ-function
KW - Minrank
KW - Orthonormal representation
KW - Turán numbers
UR - http://www.scopus.com/inward/record.url?scp=85081676053&partnerID=8YFLogxK
U2 - 10.1007/s00454-020-00185-0
DO - 10.1007/s00454-020-00185-0
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AN - SCOPUS:85081676053
SN - 0179-5376
VL - 64
SP - 654
EP - 670
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
IS - 3
ER -