On Barnette's conjecture

@article{Feder2006OnBC,
  title={On Barnette's conjecture},
  author={T. Feder and Carlos S. Subi},
  journal={Electron. Colloquium Comput. Complex.},
  year={2006}
}
Barnette’s conjecture is the statement that every 3-connected cubic planar bipartite graph is Hamiltonian. Goodey showed that the conjecture holds when all faces of the graph have either 4 or 6 sides. We generalize Goodey’s result by showing that when the faces of such a graph are 3-colored, with adjacent faces having different colors, if two of the three color classes contain only faces with either 4 or 6 sides, then the conjecture holds. More generally, we consider 3-connected cubic planar… Expand
On Barnette's conjecture
  • J. Florek
  • Mathematics, Computer Science
  • Discret. Math.
  • 2010
Barnette's conjecture is the statement that every cubic 3-connected bipartite planar graph is Hamiltonian. We show that if such a graph has a 2-factor F which consists only of facial 4-cycles, thenExpand
Thoughts on Barnette's Conjecture
TLDR
A new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian is proved, which implies the following special case of Barnette's Conjecture: if G is an Eulerian planar triangulation, whose vertices are properly coloured blue, red and green, then $G^*$ is Hamiltonian. Expand
Distance-Two Coloring of Barnette Graphs
TLDR
It is claimed that the problem remains NP-complete for tri-connected bipartite cubic planar graphs, and the problem is polynomial for cubic plane graphs with face sizes 3, 4, 5, or 6, which are called type-two Barnette graphs, because of their relation to Barnette’s second conjecture. Expand
On Barnette's Conjecture and H+-H+- property
  • J. Florek
  • Computer Science, Mathematics
  • Electron. Notes Discret. Math.
  • 2013
A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits aExpand
Distance-Two Colorings of Barnette Graphs
TLDR
It is claimed that the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which are called type-one Barnette graphs, since they are the first class identified by Barnette, and the problem is polynomial for cubic plane graphs with face sizes $3, 4, 5, $ or $6$, which are call type-two Barnettes graphs, because of their relation to Barnette's second conjecture. Expand
A Survey & Strengthening of Barnette's Conjecture
Tait and Tutte made famous conjectures stating that all members of certain graph classes contain Hamiltonian Cycles. Although the Tait and Tutte conjectures were disproved, Barnette continued thisExpand
DISTANCE-TWO COLORINGS OF CUBIC PLANAR GRAPHS
A distance-two r-coloring of a graph G is an assignment of r colors to the vertices of G so that any two vertices at distance at most two have different colors. The distance-two four-coloring problemExpand
Cuts in matchings of 3-connected cubic graphs
TLDR
A construction of 3-edge-connected digraphs satisfying the property that for every even subgraph $E$, the graph obtained by contracting the edges of $E$ is not strongly connected, disproves a recent conjecture of Hochstattler. Expand
Hamiltonian cycles in planar cubic graphs with facial 2‐factors, and a new partial solution of Barnette's Conjecture
TLDR
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. Expand
On the algorithmic complexity of finding hamiltonian cycles in special classes of planar cubic graphs
It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem evenExpand
...
1
2
...

References

SHOWING 1-10 OF 23 REFERENCES
Hamiltonicity of cubic Cayley graphs
Following a problem posed by Lov\'asz in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arisingExpand
$B$-sets and planar maps.
In this paper we examine the relation between i?-sets, which are a purely set-theoretic concept, and various concepts associated with planar maps, for instance, four-colorings, five-colorings,Expand
Convex Polytopes
Graphs, Graphs and Realizations Before proceeding to the graph version of Euler’s formula, some notation will be introduced. A (finite) abstract graph G consists of two sets, the set of vertices V =Expand
Backtrack search and look-ahead for the construction of planar cubic graphs with restricted face sizes
We describe two algorithms for the construction of simple planar cubic 3-connected graphs with all face sizes in some specified set; equivalently, simple triangulations of the plane with all vertexExpand
Hamiltonian circuits in polytopes with even sided faces
It has been conjectured that ifP is a simple 3-polytope all of whose faces have an even number of sides, thenP has a Hamiltonian circuit. In this paper it is shown that, if all the faces ofP areExpand
Matroid matching and some applications
  • L. Lovász
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 1980
Abstract The polymatroid matching problem, also known as the matchoid problem or the matroid parity problem, is polynomially unsolvable in general but solvable for linear matroids. The solution forExpand
Efficient Algorithms for Graphic Matroid Intersection and Parity (Extended Abstract)
TLDR
Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network. Expand
Hamiltonicity of planar cubic multigraphs
  • Z. Skupien
  • Mathematics, Computer Science
  • Discret. Math.
  • 2002
Hamiltonicity of connected cubic planar general graphs G is characterized in terms of partitioning any dual graph G* into two trees. Thus tree-tree triangulations become involved. The related SteinExpand
The Planar Hamiltonian Circuit Problem is NP-Complete
TLDR
The problem of determining whether a planar, cubic, triply-connected graph G has a Hamiltonian circuit is considered and it is shown that this problem is NP-complete. Expand
A class of Hamiltonian polytopes
  • P. Goodey
  • Mathematics, Computer Science
  • J. Graph Theory
  • 1977
It is shown that any simple 3-polytope, all of whose faces are triangles or hexagons, admits a hamiltonian circuit.
...
1
2
3
...