The maximum spectral radius of outerplanar and planar $k$-uniform hypergraphs

Abstract: For an integer $k\ge3$, a $k$-angulation is a simple $2$-connected outerplane graph whose interior faces are bounded by $k$-cycles, and a closed $k$-angulation is a simple $2$-connected plane graph all of whose faces, the outer face included, are bounded by $k$-cycles; the face hypergraph of either is the $k$-uniform hypergraph whose edges are the vertex sets of those faces. For $k=3$ these are the outerplanar and planar hypergraphs of Ellingham, Lu and Wang, who determined the outerplanar extremal hypergraph for large $n$ and conjectured the planar one. In this paper, we determine the extremal hypergraphs in both classes for every $k$. In the outerplanar case, for all sufficiently large admissible $n$, it is the fan, in which a single vertex lies on every face, and the maximum equals $(4f)<sup>{1/k}(1+o(1))$ with $f=(n-2)/(k-2)$. In the planar problem the maximum has order $n<sup>{1/3}$ when $k=3$ and order $n<sup>{2/k}$ when $k\ge4$. For $k\ge4$ the extremal hypergraphs are the face hypergraphs of the balanced theta graphs, in which two vertices are joined by internally disjoint paths and every face is a $k$-cycle through both: for $k=4$, where the closed $4$-angulations are the quadrangulations of the sphere, this holds for every $n\ge5$, the extremal hypergraph being $\mathcal{H}(K_{2,n-2})$, and for $k\ge5$ for all sufficiently large admissible $n$. For $k\ge6$ the extremal hypergraph is not unique: when the number of faces is even there are exactly $\lfloor(k-2)/2\rfloor$ of them up to isomorphism. For $k=3$ two vertices of a plane triangulation lie on at most two common faces, the balanced theta graphs are unavailable, and the extremal hypergraph is instead, for all sufficiently large $n$, the face hypergraph of $K_2+P_{n-2}$; this confirms a conjecture of Ellingham, Lu and Wang.