On Barnette’s conjecture

Abstract

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 a partition of its vertex set into two subsets so that each induces a tree.Let G be a simple even plane triangulation and suppose that V 1 , V 2 , V 3 is a 3-coloring of the vertex set of G. Let B i , i = 1, 2, 3, be the set of all vertices in V i of the degree at least 6. We prove that if induced graphs G[B 1 ∪ B 2 ] and G[B 1 ∪ B 3 ] are acyclic, then the following properties are satisfied:

Remarks on Barnette’s conjecture

“…. f k in P is called a sequence from f 0 to f k with length k. The distance of two faces f, h in P is the length of a shortest sequence of faces from f to h. We prove the following theorem which generalizes results obtained by Florek in [3]. Theorem 1.2.…”

“…is a vertex partition of V (α(G)) into independent colour classes such that α(R) contains only vertices of degree 4. It was proved in [3] that every graph G ∈ E(4) can be constructed (up to isomorphism) from the octahedron by iterating the operation α. Hence, by induction we obtain: (i) |B ∪ W | > 1 2 |G|.…”

Features of Identification Elements Deployed along Highways: Example of Ukraine and India

“…The place identification is an important area of research in various fields of science (Florek, 2010;Morgan, 2010;Scannell and Gifford, 2010). These planning theories are relevant for spaces along highways, because here every element affects the full perception of the place.…”

Hamiltonian cycles in some family of cubic 3-connected plane graphs

“…(ω(z(A(G n ))), g), forz ∈ Φ Z (n) andω ∈ Ω * (z(A(G n )), z(Z n )), or (H n , g), for n ≡ 1, 2, 3 (mod 5). From (7), it follows that every graph (G, g) ∈ S n , for n > 10, is also opequivalent to one of the above graphs (or to a mirror reflection of one of them, for n odd). Finally, by (8) and Lemma 3.2, every graph (G, g) ∈ S n , different than (H 7 , g), has a hamiltonian set which is (±)compatible with the orientation of c(G).…”

“…Thomassen [17] found a shorter proof of this theorem. It is known (see Stein [16], Florek [7], Alt, Payne, Schmidt and Wood [2]) that the dual version of the Barnette's conjecture is the following: Every plane triangulation with all vertices of degree at most 6 has an induced tree with the vertex set dominating all faces of the graph. Let G be the set of all plane triangulations (G, g) such that every vertex of G different than g has degree at most 6, and every vertex adjacent to g has degree at most 5.…”