TY - JOUR
T1 - There is no McLaughlin geometry
AU - Östergård, Patric R.J.
AU - Soicher, Leonard H.
PY - 2018/4/1
Y1 - 2018/4/1
N2 - We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.
AB - We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.
KW - Backtrack search
KW - Krein bound
KW - McLaughlin geometry
KW - McLaughlin graph
KW - Partial geometry
KW - Pseudogeometric graph
UR - http://www.scopus.com/inward/record.url?scp=85033579374&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2017.10.004
DO - 10.1016/j.jcta.2017.10.004
M3 - Article
AN - SCOPUS:85033579374
VL - 155
SP - 27
EP - 41
JO - Journal of Combinatorial Theory Series A
JF - Journal of Combinatorial Theory Series A
SN - 0097-3165
ER -