A condition for a graph in the disc to contain the half hemisphere of the dodecahedron -- application to Dehn surgery theory (Q2780946)

This article introduces two results on graph theory which were applied in the previous paper of the authors [Math. Proc. Camb. Philos. Soc. 123, No, 3, 501-529 (1998; Zbl 0910.57005)] to prove that any Dehn surgery on a non-cabled symmetric knot does not yield a reducible 3-manifold.NEWLINENEWLINENEWLINEThe first result is the following. Let \(i=3,4\) or 5, then a finite graph \(\Lambda\) embedded in a 2-sphere contains the hemisphere of the tetrahedron, hexahedron or dodecahedron, according to \(i=3,4\) or 5, if \(\Lambda\) is a graph satisfying the following conditions: (i) \(\Lambda\) does not have a cut-vertex; (ii) all the faces of \(\Lambda\) are \(i\)-gons, but only one face is a \(q\)-gon \((q\geq 2)\); (iii) all the vertices of \(\Lambda\) are of valency 3, except for at most \(i-1\) singular vertices of valencies not less than 2; (iv) the singular vertices belong to the \(q\)-gon. Moreover, a pattern of the hemisphere can be found such that it does not contain the \(q\)-gon and its vertices of valency 3 are not singular. The proof of these results was given in case of \(i=5\) in the previously cited paper, and in case of \(i=4\) in this article. The proof is trivial in case of \(i=3\).NEWLINENEWLINENEWLINEThe second result is the following. Let \(\Lambda\) be a graph embedded in a disc \(D^2\) such that the boundary \(\partial D^2\) is a union of edges of \(\Lambda\). Assume that the vertices on \(\partial D^2\) are of valency 3 and the other vertices of \(\Lambda\) are of valency 4. Suppose that each edge of \(\Lambda\) is labelled by \(\alpha\) or \(\beta\) in such a way that every pair of opposite edges at each vertex of valency 4 is separated into one with a label \(\alpha\) and the other with a label \(\beta\) and so that every pair of adjacent edges on \(\partial D^2\) are separated similarly. Then there is a face of \(\Lambda\) such that it is incident either only to edges with label \(\alpha\) or only to those with label \(\beta\). This result had been already proved for the hemisphere of the dodecahedron in the previously cited paper. But the proof given in this article reveals that the above fact is rather trivial.