外平面与平面$k$-一致超图的最大谱半径

中文摘要 AI 辅助

对于整数$k\ge3$，一个$k$-角化是一个简单$2$-连通的外平面图，其内部面由$k$-圈围成，而一个闭$k$-角化是一个简单$2$-连通的平面图，其所有面（包括外部面）都由$k$-圈围成；两者的面超图是$k$-一致超图，其边是这些面的顶点集。对于$k=3$，这些是Ellingham、Lu和Wang的外平面和平面超图，他们确定了对于大的$n$的外平面极值超图，并猜想平面极值超图。在本文中，我们确定了对于每个$k$的两个类别中的极值超图。在外平面情形中，对于所有足够大的可行$n$，极值超图是扇，其中单个顶点位于每个面上，最大值等于$(4f)^{1/k}(1+o(1))$，其中$f=(n-2)/(k-2)$。在平面问题中，当$k=3$时最大值阶为$n^{1/3}$，当$k\ge4$时阶为$n^{2/k}$。对于$k\ge4$，极值超图是平衡theta图的面超图，其中两个顶点由内部不相交的路径连接，且每个面都是通过这两个顶点的$k$-圈：对于$k=4$，闭$4$-角化是球面的四边形剖分，这适用于每个$n\ge5$，极值超图为$\mathcal{H}(K_{2,n-2})$，对于$k\ge5$适用于所有足够大的可行$n$。对于$k\ge6$，极值超图不唯一：当面数为偶数时，同构意义下恰好有$\lfloor(k-2)/2\rfloor$个。对于$k=3$，平面三角剖分的两个顶点至多位于两个共同面上，平衡theta图不可用，极值超图改为对于所有足够大的$n$，是$K_2+P_{n-2}$的面超图；这证实了Ellingham、Lu和Wang的一个猜想。

英文摘要

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 $(4f)^{1/k}(1+o(1))$ with $f=(n-2)/(k-2)$. In the planar problem the maximum has order $n^{1/3}$ when $k=3$ and order $n^{2/k}$ 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.

发表机构

• Sungkyunkwan University（成均馆大学）
• The State University of New York, Korea（纽约州立大学韩国分校）

机构由 AI 辅助整理，请以论文原文为准。

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50 pages