Locally Hamiltonian graphs and minimal size of maximal graphs on a surface

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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