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Hamiltonian cycles in planar cubic graphs with facial 2‐factors, and a new partial solution of Barnette's Conjecture
Author(s) -
Bagheri Gh Behrooz,
Feder Tomas,
Fleischner Herbert,
Subi Carlos
Publication year - 2021
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22612
Subject(s) - combinatorics , mathematics , bipartite graph , cubic graph , planar graph , polyhedral graph , conjecture , hamiltonian path , discrete mathematics , spanning tree , foster graph , 1 planar graph , graph , chordal graph , voltage graph , line graph
We study the existence of hamiltonian cycles in plane cubic graphs G having a facial 2‐factor Q . Thus hamiltonicity in G is transformed into the existence of a (quasi) spanning tree of faces in the contraction G ∕ Q . 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.