局部哈密顿图与曲面上极大图的最小尺寸
Locally Hamiltonian graphs and minimal size of maximal graphs on a surface
摘要：
我们证明了每一个具有$n\ge 3$顶点和可能具有重边的局部哈密顿图至少有$3n-6$条边相等当且仅当它三角化球面。因此，图$G$图在某个二维曲面$Σ$（不一定是紧的）上的每一个边极大嵌入至少有$3n-6$相等的边当且仅当$G$也对球面进行三角剖分。此外，如果$G$是简单的，则对于每个顶点$v$，$Σ$上围绕$v$的边的循环排序与球面上围绕$v$的顺时针或逆时针方向相同。如果$G$不包含4个顶点的完全图，并且至少有4个顶点，则两个嵌入的面边界相同。
英文摘要：
We prove that every locally Hamiltonian graph with $n\ge 3$ vertices and possibly with multiple edges has at least $3n-6$ edges with equality if and only if it triangulates the sphere. As a consequence, every edge-maximal embedding of a graph $G$ graph on some 2-dimensional surface $Σ$ (not necessarily compact) has at least $3n-6$ edges with equality if and only if $G$ also triangulates the sphere. If, in addition, $G$ is simple, then for each vertex $v$, the cyclic ordering of the edges around $v$ on $Σ$ is the same as the clockwise or anti-clockwise orientation around $v$ on the sphere. If $G$ contains no complete graph on 4 vertices and has at least 4 vertices, then the face-boundaries are the same in the two embeddings.
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