Universal graphs and universal permutations

Abstract: Let $X$ be a family of graphs and $X_n$ the set of $n$-vertex graphs in $X$. A graph $U^{(n)}$ containing all graphs from $X_n$ as induced subgraphs is called $n$-universal for $X$. Moreover, we say that $U^{(n)}$ is a proper $n$-universal graph for $X$ if it belongs to $X$. In the present paper, we construct a proper $n$-universal graph for the class of split permutation graphs. Our solution includes two ingredients: a proper universal 321-avoiding permutation and a bijection between 321-avoiding permutations and symmetric split permutation graphs. The $n$-universal split permutation graph constructed in this paper has $4n^3$ vertices, which means that this construction is order-optimal.