We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2-factor . Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction ∕ G . In particular, we study the case where G is the leapfrog extension (called vertex envelope of a plane cubic graph G 0. As a consequence we prove hamiltonicity in the leapfrog extension of planar cubic cyclically 4-edge-connected bipartite graphs. This and other results of this paper establish partial solutions of Barnette's Conjecture according to which every 3-connected cubic planar bipartite graph is hamiltonian. These results go considerably beyond Goodey's result on this topic.

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Let [Formula: see text] and [Formula: see text] be two labeled graphs of order [Formula: see text]. For any permutation [Formula: see text] the [Formula: see text]-permutation graph of labeled graphs [Formula: see text] and [Formula: see text] is the union of [Formula: see text] and [Formula: see text] together with the edges joining the vertex [Formula: see text] to the vertex [Formula: see text]. This operation on graphs is useful to produce a large class of networks with approximately the same properties as one of the original networks or even smaller. In this work we consider some properties of the permutation graph [Formula: see text], for labeled graph [Formula: see text] and [Formula: see text] of the same order. We provide bounds for the parameters radius, diameter, total distance, connectivity, edge-connectivity, chromatic number, and edge-chromatic number.

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