Extension of a theorem of Whitney

It is shown that every planar graph with no separating triangles is a subgraph of a Hamiltonian planar graph; that is, Whitney’s theorem holds without the assumption of a triangulation.     

Each maximal planar graph with exactly two separating triangles is Hamiltonian

A classical result of Whitney states that each maximal planar graph without separating triangles is Hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Chen [Any maximal planar graph with only one separating triangle is Hamiltonian J. Combin. Optim. 7 (2003) 79-86] proved that any maximal planar graph with only one separating triangle is still Hamiltonian. In this paper, it is shown that the conclusion of Whitney's Theorem still holds if there are exactly two separating triangles.     

Hamilton cycles in plane triangulations

We extend Whitney's Theorem that every plane triangulation without separating triangles is hamiltonian by allowing some separating triangles. More precisely, we define a decomposition of a plane triangulation G into 4‐connected ‘pieces,’ and show that if each piece shares a triangle with at most three other pieces then G is hamiltonian. We provide an example to show that our hypothesis that each piece shares a triangle with at most three other pieces' cannot be weakened to ‘four other pieces.’ As part of our proof, we also obtain new results on Tutte cycles through specified vertices in planar graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 138–150, 2002     

Concept of a vertex in a matroid and 3-connected graphs

The concept of a matroid vertex is introduced. The vertices of a matroid of a 3-connected graph are in one-to-one correspondence with vertices of the graph. Thence directly follows Whitney's theorem that cyclic isomorphism of 3-connected graphs implies isomorphism. The concept of a vertex of a matroid leads to an equally simple proof of Whitney's theorem on the unique embedding of a 3-connected planar graph in the sphere. It also leads to a number of new facts about 3-connected graphs. Thus, consideration of a vertex in a matroid that is the dual of the matroid of a graph leads to a natural concept of a nonseparating cycle of a graph. Whitney's theorem on cyclic isomorphism can be strengthened (even if the nonseparating cycles of a graph are considered, the theorem is found to work) and a new criterion for planarity of 3-connected graphs is obtained (in terms of nonseparating cycles).     

On the tightness conditions for maximal planar graphs

We characterize the tight structure of a vertex-accumulation-free maximal planar graph with no separating triangles. Together with the result of Halin who gave an equivalent form for such graphs, this yields that a tight structure always exists in every 4-connected maximal planar graph with one end.     

On Hamilton cycles in certain planar graphs

Let G be a 2-connected plane graph with outer cycle X_G such that for every minimal vertex cut S of G with |S| ≤ 3, every component of G\S contains a vertex of X_G. A sufficient condition for G to be Hamiltonian is presented. This theorem generalizes both Tutte's theorem that every 4-connected planar graph is Hamiltonian, as well as a recent theorem of Dillencourt about NST-triangulations. A linear algorithm to find a Hamilton cycle can be extracted from the proof. One corollary is that a 4-connected planar graph with the vertices of a triangle deleted is Hamiltonian. © 1996 John Wiley & Sons, Inc.     

Vertex 2-isomorphism

Whitney's theorem on 2-isomorphism characterizes the set of graphs having the same cycles as a given graph, where a cycle is regarded as a set of edges. In this paper, vertex 2-isomorphism is defined and used to prove a vertex analogue of Whitney's theorem. The main theorem states that two connected graphs have the same set of cycles, where a cycle is now regarded as a set of vertices, if and only if one can be obtained from the other by a sequence of simple operations. © 1992 John Wiley & Sons, Inc.     

Generalized delaunay triangulation for planar graphs

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graphG=(V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs inO(|V|²) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-and-conquer algorithm based on the polygon cutting theorem of Chazelle is given that runs inO(|V| log |V|) time.Supported in part by the National Science Foundation under Grants DCR 8420814 and ECS 8340031.     

A note on Barnette’s conjecture

A well-known conjecture of Barnette states that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle, which is equivalent to the statement that every 3-connected even plane triangulation admits a 2-tree coloring, meaning that the vertices of the graph have a 2-coloring such that each color class induces a tree. In this paper we present a new approach to Barnette’s conjecture by using 2-tree coloring.A Barnette triangulation is a 3-connected even plane triangulation, and a B-graph is a smallest Barnette triangulation without a 2-tree coloring. A configuration is reducible if it cannot be a configuration of a B-graph. We prove that certain configurations are reducible. We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette’s conjecture.     

On the Strongest Form of a Theorem of Whitney for Hamiltonian Cycles in Plane Triangulations

In this article, we investigate hamiltonian cycles in plane triangulations. The aim of the article is to find the strongest possible form of Whitney's theorem about hamiltonian triangulations in terms of the decomposition tree defined by separating triangles. We will decide on the existence of nonhamiltonian triangulations with given decomposition trees for all trees except trees with exactly one vertex with degree and all other degrees at most 3. For these cases, we show that it is sufficient to decide on the existence of nonhamiltonian triangulations with decomposition tree K_{1, 4} or K_{1, 5}. We also give computational results on the size of a possible minimal nonhamiltonian triangulation with these decomposition trees.     

平面图3色可染的一个充分条件

Steinberg猜想既没有4-圈又没有5-圈的平面图是3色可染的. Xu, Borodin等人各自独立地证明了既没有相邻三角形又没有5-和7-圈的平面图是3 色可染的. 作为这一结果的推论, 没有4-, 5-和7-圈的平面图是3色可染的. 本文证明一个比此推论更接近Steinberg猜想的结果, 设G是一个既没有4-圈又没有5-圈的平面图, 若对每一个k∈{3, 6, 7}, G都不含(k, 7)-弦, 则G是3色可染的, 这里的(k, 7)-弦是指长度为7+k-2的圈的一条弦, 它的两个端点将圈分成两条路, 一条路的长度为6, 另一条路的长度为k-1.     

One-way infinite hamiltonian paths in infinite maximal planar graphs

A one-way infinite Hamiltonian path is constructed in an infinite 4-connected VAP-free maximal planar graph containing one or two vertices of infinite degree. Combining this result and that of R. HALIN who investigated the structure of such graphs, we conclude that such a path always exists in every infinite 4-connected maximal planar graph with exactly one end, which is an extension of H. WHITNEY'S theorem to infinite graphs.     

关于平面图(3,1)~*-可选性的一个注记(英文)

给出了平面图的一个结构性定理,并证明了每个没有5-圈,相邻三角形,相邻四边形的平面图是(3,1)*-可选色的.     

关于平面图（3，1）＊-可选性的一个注记

给出了平面图的一个结构性定理,并证明了每个没有5-圈,相邻三角形,相邻四边形的平面图是（3,1）＊-可选色的．     

Non-hamiltonian triangulations with distant separating triangles

In 1996 Böhme, Harant, and Tká? asked whether there exists a non-hamiltonian triangulation with the property that any two of its separating triangles lie at distance at least 1. Two years later, Böhme and Harant answered this in the affirmative, showing that for any non-negative integer $d$ there exists a non-hamiltonian triangulation with seven separating triangles every two of which lie at distance at least $d$. In this note we prove that the result holds if we replace seven with six, remarking that no non-hamiltonian triangulation with fewer than six separating triangles is known.     

On the number of hamiltonian cycles in a maximal planar graph

We consider the problem of the minimum number of Hamiltonian cycles that could be present in a Hamiltonian maximal planar graph on p vertices. In particular, we construct a p-vertex maximal planar graph containing exactly four Hamiltonian cycles for every p ≥ 12. We also prove that every 4-connected maximal planar graph on p vertices contains at least p/(log₂ p) Hamiltonian cycles.     

Orienting planar graphs

It is shown that every maximal planar graph (triangulation) can be contracted at an arbitrary point (by identifying it with an adjacent point) so that triangularity is preserved. This is used as a lemma to prove that every triangulation can be (a) oriented so that with three exceptions every point has indegree three, the exceptions having indegrees 0, 1 and 2, and (b) decomposed into three edge-disjoint trees of equal order. The lemma also provides an elementary proof of a theorem of Wagner that every triangulation can be represented by a straight-line drawing.     

On the number of Hamiltonian cycles in triangulations

It is proved that if a planar triangulation different from K₃ and K₄ contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [2], this yields that, for n ? 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3-connected triangulated graphs.     

Triangulations intersect nicely

We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.     

A theorem on graph embedding with a relation to hyperbolic volume

We prove that a planar cubic cyclically 4-connected graph of odd χ < 0 is the dual of a 1-vertex triangulation of a closed orientable surface. We explain how this result is related to (and applied to prove at a separate place) a theorem about hyperbolic volume of links: the maximal volume of alternating links of given χ < 0 does not depend on the number of their components.