11 problems

Let $G$) be a transitive permutation group of degree $n$, and let $\Gamma_G$ denote its derangement graph. For a positive integer $c$, say that $\Gamma_G$ contains no clique of siz…

Meagher–Razafimahatratra–Spiga's conjecture. There exists a transitive permutation group $G$ of degree $n$ such that $\Gamma_G$ is a complete multipartite graph with $n/2$ parts.

Maximum intersection-density characterization conjecture. The only transitive groups that attain the upper bound $n/3$ are those whose derangement graphs are complete tripartite gr…

Multipartite derangement graph conjecture. If $n$ is even but not a power of $2$, then there is a transitive group $G$ of degree $n$ such that its derangement graph is a complete m…

Conjectures for $I(n)$. 1. If $n$ is even but not a power of $2$, then there is a transitive group $G$ of degree $n$ such that $\Gamma_G$ is a complete multipartite graph with…

Tripartite classification conjecture. If

Let $G$ be a $3$-transitive permutation group of degree $n$ acting on a set $\Omega$, and let $\Gamma_G$ be its derangement graph. An independent set is a set of vertices containin…

Let $\mathbb{P}_q^n$ be projective $n$-space, with $n\geq 1$, and let $\Gamma_{\operatorname{PGL}(n+1,q)}$ be the derangement graph of $\operatorname{PGL}(n+1,q)$ acting on…

Let $\mathbb{P}_q$ be the projective line and let $\Gamma_{\operatorname{PSL}(2,q)}$ be the derangement graph of $\operatorname{PSL}(2,q)$ acting on it. An independent set is a set…

Lexicographic eigenvalue-ordering conjecture. For every partition $\lambda=(\lambda_1,\ldots,\lambda_s)\vdash n$,

Ku–Wong's conjecture. The smallest eigenvalue of $\Gamma_n$ is