Even cycles with prescribed chords in planar cubic graphs

AbstractThe following result is being proved. Theorem: Let e be an arbitrary line of the 2-connected, 3-regular, planar graph G such that e does not belong to any cut set of size 2. The G contains an even cycle for which e is a chord

A note about the dominating circuit conjecture

AbstractThe dominating circuit conjecture states that every cyclically 4-edge-connected cubic graph has a dominating circuit. We show that this is equivalent to the statement that any two edges of such a cyclically 4-edge-connected graph are contained in a dominating circuit

On circuit decomposition of planar Eulerian graphs

AbstractWe give a common generalization of P. Seymour's “Integer sum of circuits” theorem and the first author's theorem on decomposition of planar Eulerian graphs into circuits without forbidden transitions

A universal set of growth operations for fullerenes

We give a simple set of growth operations that suffice to generate all fullerene isomer structures

On finding hamiltonian cycles in Barnette graphs

In this paper, we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite graphs is an NP-complete problem.Comment: arXiv admin note: substantial text overlap with arXiv:1806.0671