Maximal planar graphs of inscribable type and diagonal flips

A theorem due to Wagner states that given two maximal planar graphs with n vertices, one can be obtained from the other by performing a finite sequence of diagonal flips. In this paper, we show a result of a similar flavour—given two maximal planar graphs of inscribable type having the same vertex set, one can be obtained from the other by performing a finite sequence of diagonal flips such that all the intermediate graphs are of inscribable type.     

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

In 1956, W.T. Tutte proved that every 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. It is shown that Sanders’ result is best possible by constructing 4-connected maximal planar graphs with three edges a large distance apart such that any hamiltonian cycle misses one of them. If the maximal planar graph is 5-connected then such a construction is impossible.     

On certain Hamiltonian cycles in planar graphs

The problem is considered under which conditions a 4-connected planar or projective planar graph has a Hamiltonian cycle containing certain prescribed edges and missing certain forbidden edges. The results are applied to obtain novel lower bounds on the number of distinct Hamiltonian cycles that must be present in a 5-connected graph that is embedded into the plane or into the projective plane with face-width at least five. Especially, we show that every 5-connected plane or projective plane triangulation on n vertices with no non-contractible cyles of length less than five contains at least distinct Hamiltonian cycles. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 81–96, 1999     

Packing trees into planar graphs

In this study, we provide methods for drawing a tree with n vertices on a convex polygon, without crossings and using the minimum number of edges of the polygon. We apply the results to obtain planar packings of two trees in some specific cases. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 172–181, 2002     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

Extending colorings of locally planar graphs

Suppose G is a graph embedded in S_g with width (also known as edge width) at least 264(2^g−1). If P ⊆ V(G) is such that the distance between any two vertices in P is at least 16, then any 5‐coloring of P extends to a 5‐coloring of all of G. We present similar extension theorems for 6‐ and 7‐chromatic toroidal graphs, for 3‐colorable large‐width graphs embedded on S_g with every face even‐sided, and for 4‐colorable large‐width Eulerian triangulations. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 105–116, 2001     

Star chromatic numbers of some planar graphs

The concept of the star chromatic number of a graph was introduced by Vince (A. Vince, Star chromatic number, J. Graph Theory 12 (1988), 551–559), which is a natural generalization of the chromatic number of a graph. This paper calculates the star chromatic numbers of three infinite families of planar graphs. More precisely, the first family of planar graphs has star chromatic numbers consisting of two alternating infinite decreasing sequences between 3 and 4; the second family of planar graphs has star chromatic numbers forming an infinite decreasing sequence between 3 and 4; and the third family of planar graphs has star chromatic number 7/2. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 33–42, 1998     

On 2-walks in chordal planar graphs

A 2-walk is a closed spanning trail which uses every vertex at most twice. A graph is said to be chordal if each cycle different from a 3-cycle has a chord. We prove that every chordal planar graph G with toughness has a 2-walk.     

Combinatorial curvature for planar graphs

Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvature of a manifold. We show this curvature has the property that its negative values are bounded above by a universal negative constant. We also prove that G is hyperbolic if its curvature is negative. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 220–229, 2001     

Homomorphism bounds for oriented planar graphs

If is a class of oriented graphs (directed graphs without opposite arcs), then an oriented graph is a homomorphism bound for if there is a homomorphism from each graph in to H. We find some necessary conditions for a graph to be a homomorphism bound for the class of oriented planar graphs and prove that such a graph must have maximum degree at least 16; thus there exists an oriented planar graph with oriented chromatic number at least 17. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 175–190, 2007     

平面图的强边染色

图$G$的强边染色是在正常边染色的基础上, 要求距离不超过$2$的任意两条边染不同的颜色, 强边染色所用颜色的最小整数称为图$G$的强边色数。本文首先给出极小反例的构型, 然后通过权转移法, 证明了$g(G)\geq5$, $\Delta(G)\geq6$且$5$-圈不相交的平面图的强边色数至多是$4\Delta(G)-1$。     

平面图的强边染色

图$G$的强边染色是在正常边染色的基础上, 要求距离不超过$2$的任意两条边染不同的颜色, 强边染色所用颜色的最小整数称为图$G$的强边色数。本文首先给出极小反例的构型, 然后通过权转移法, 证明了$g(G)\geq5$, $\Delta(G)\geq6$且$5$-圈不相交的平面图的强边色数至多是$4\Delta(G)-1$。     

One‐way infinite 2‐walks in planar graphs

We prove that every 3‐connected 2‐indivisible infinite planar graph has a 1‐way infinite 2‐walk. (A graph is 2‐indivisible if deleting finitely many vertices leaves at most one infinite component, and a 2‐walk is a spanning walk using every vertex at most twice.) This improves a result of Timar, which assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also provide other results when the graph is bipartite or an infinite analog of a triangulation: then the prism over the graph has a spanning 1‐way infinite path.     

Cubic vertices in planar hypohamiltonian graphs

Thomassen showed in 1978 that every planar hypohamiltonian graph contains a cubic vertex. Equivalently, a planar graph with minimum degree at least 4 in which every vertex-deleted subgraph is hamiltonian, must be itself hamiltonian. By applying work of Brinkmann and the author, we extend this result in three directions. We prove that (i) every planar hypohamiltonian graph contains at least four cubic vertices, (ii) every planar almost hypohamiltonian graph contains a cubic vertex, which is not the exceptional vertex (solving a problem of the author raised in J. Graph Theory [79 (2015) 63–81]), and (iii) every hypohamiltonian graph with crossing number 1 contains a cubic vertex. Furthermore, we settle a recent question of Thomassen by proving that asymptotically the ratio of the minimum number of cubic vertices to the order of a planar hypohamiltonian graph vanishes.     

Triangle-free planar graphs with the smallest independence number

Steinberg and Tovey proved that every -vertex planar triangle-free graph has an independent set of size at least , and described an infinite class of tight examples. We show that all -vertex planar triangle-free graphs except for this one infinite class have independent sets of size at least .     

Gk^d图的平面性和平面图的色数

利用欧拉公式研究了Gdk图的平面性,获得了一个重要定理,并由此得到了关于平面图色数的一个结论.     

Hamiltonicity of planar graphs with a forbidden minor

Tutte showed that -connected planar graphs are Hamiltonian, but it is well known that -connected planar graphs need not be Hamiltonian. We show that -minor-free -connected planar graphs are Hamiltonian. This does not extend to -minor-free -connected graphs in general, as shown by the Petersen graph, and does not extend to -minor-free -connected planar graphs, as we show by an infinite family of examples.