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“…For every graph G in G, there is a sequence G 0 , G 1 , ..., G t of members of G such that G 0 is the tetrahedron, G t is G, and each G i+1 is obtained from G i by adding a handle. In [1,2,4,12], analogous results are obtained for 3-regular simple planar graphs with other connectivities. For 4-regular simple planar graphs, the situation is similar and the readers are referred to [3,9,10].…”

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“…For every graph G in G, there is a sequence G 0 , G 1 , ..., G t of members of G such that G 0 is the tetrahedron, G t is G, and each G i+1 is obtained from G i by adding a handle. In [1,2,4,12], analogous results are obtained for 3-regular simple planar graphs with other connectivities. For 4-regular simple planar graphs, the situation is similar and the readers are referred to [3,9,10].…”

“…Since v 4 is in the exterior of Δ 1 and v 3 is in the exterior of Δ 2 , the convex hull ch(v 1 4 ) in counterclockwise order. By Lemma 4, T 1 (resp., T 2 ) has a flippable edge e 1 (resp., e 2 ) in a triangle opposite to s. If flipping edge e 1 in T 1 or edge e 2 in T 2 increases the degree of s to 5, then perform the edge flip.…”

“…We have now generated all C5CPs on up Downloaded 05/11/18 to 34.218.44.141. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php to 52 vertices using the generation procedure of Barnette [1] and Butler [6]. The graphs were then checked for hamiltonian cycles.…”