TY - JOUR
T1 - Turán, involution and shifting
AU - Kalai, Gil
AU - Nevo, Eran
N1 - Funding Information:
Acknowledgements. Research of Kalai is partially supported by ERC advanced grant 320924, BSF grant 2006066, and NSF grant DMS-1300120, and of Nevo by Israel Science Foundation grant ISF-1695/15, by grant 2528/16 of the ISF-NRF Singapore joint research program, and by ISF-BSF joint grant 2016288.
Publisher Copyright:
© The journal and the authors, 2019.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - We propose a strengthening of the conclusion in Turán’s (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel–Turán theorem by weakening its assumption: for any graph G on n vertices and any involution on its vertex set, if for any 3-set S of the vertices, the number of edges in G spanned by S, plus the number of edges in G spanned by the image of S under the involution, is at least 2, then the number of edges in G is at least the Mantel–Turán bound, namely the number achieved by two disjoint cliques of sizes n2 rounded up and down.
AB - We propose a strengthening of the conclusion in Turán’s (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel–Turán theorem by weakening its assumption: for any graph G on n vertices and any involution on its vertex set, if for any 3-set S of the vertices, the number of edges in G spanned by S, plus the number of edges in G spanned by the image of S under the involution, is at least 2, then the number of edges in G is at least the Mantel–Turán bound, namely the number achieved by two disjoint cliques of sizes n2 rounded up and down.
KW - Shifting
KW - Threshold graphs
KW - Turán’s (3,4)-conjecture
UR - http://www.scopus.com/inward/record.url?scp=85111608738&partnerID=8YFLogxK
U2 - 10.5802/ALCO.30
DO - 10.5802/ALCO.30
M3 - Article
AN - SCOPUS:85111608738
SN - 2589-5486
VL - 2
SP - 367
EP - 378
JO - Algebraic Combinatorics
JF - Algebraic Combinatorics
IS - 3
ER -