Kourovka Notebook Problem 12.59

Definitions and mathematical statement

A strongly regular graph with parameters $(85,14,3,2)$ means a finite simple undirected graph $G=(V,E)$ such that $|V|=85$, every vertex has exactly $14$ neighbors, every two distinct adjacent vertices have exactly $3$ common neighbors, and every two distinct nonadjacent vertices have exactly $2$ common neighbors. For $v\in V$, write $N_G(v)=\{u\in V:\{u,v\}\in E\}$; the neighborhood graph of $v$ is the induced graph $G[N_G(v)]$.

Does there exist such a graph $G$ and a vertex $v\in V$ for which $G[N_G(v)]$ is disconnected? Explicitly, the additional requirement is that there exist nonempty disjoint sets $A,B\subseteq N_G(v)$ with $N_G(v)=A\cup B$ and $\{a,b\}\notin E$ for every $a\in A$ and $b\in B$.

Short English statement

Does there exist a strongly regular graph with parameters $(85,14,3,2)$ in which the subgraph induced by the neighbors of at least one vertex is disconnected?

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $700878$, exact source problem code catalogue entry, titled Kourovka Notebook Problem 12.59.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 12.59, 12th issue (1992); proposed by A. A. Makhnëv.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. This bibliographic attribution is reproduced from the record and has not been independently verified.