Zhicheng Gao

The Distribution of the Maximum Vertex Degree in Random Planar Maps

We determine the limiting distribution of the maximum vertex degree ?n in a random triangulation of an n-gon, and show that it is the same as that of the maximum of n independent identically distributed random variables G2, where G2 is the sum of two independent geometric(1/2) random variables. This answers affirmatively a question of Devroye, Flajolet, Hurtado, Noy and Steiger, who gave much weaker almost sure bounds on ?n. An interesting consequence of this is that the asymptotic probability that a random triangulation has a unique vertex with maximum degree is about 0.72. We also give an analogous result for random planar maps in general.

2-walks in circuit graphs

Abstract We prove the conjecture of Jackson and Wormald that every 3-connected planar graph has a closed walk visiting every vertex once or twice. This strengthens Barnette′s Theorem that every 3-connected planar graph has a spanning tree with maximum degree at most 3. The result also holds for 3-connected projective planar graphs.

The numbers of rooted triangular maps on a surface

Abstract In this paper we enumerate the rooted triangular maps on a surface by number of vertices. A parametric expression of the generating function is obtained for such maps on the sphere, the projective plane, and the torus. An asymptotic formula is obtained for such maps on an arbitrary surface.

The Distribution of Heights of Binary Trees and Other Simple Trees

The number of binary trees of fixed size and given height is estimated asymptotically near the peak of the distribution. There, a local limit theorem with convergence to a theta law is established. Large deviation bounds corresponding to large heights and small heights are also derived. The methods based on the analysis of singular iterations apply to any simple family of trees.

The Size of the Largest Components in Random Planar Maps

Bender, Richmond, and Wormald showed that in almost all planar 3-connected triangulations (or dually, 3-connected cubic maps) with n edges, the largest 4-connected triangulation (or dually, the largest cyclically 4-edge-connected cubic component) has about n/2 edges [ Random Structures Algorithms, 7 (1995), pp. 273--285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps.

Submaps of maps. I: General 0–1 laws

Abstract Let M n be the set of n edge maps of some class on a surface of genus g. When g = 0 (planar maps) we show how to prove that limn → ∞ | M n|1/n exists for many classes of maps. Let P be a particular map that can appear as a submap of maps in our class. There is often a strong 0–1 law for the property that P is a submap of a randomly chosen map in M n: If P is planar, then almost all M n contain at least cn disjoint copies of P for small enough c; while if P is not planar, almost no M n contain a copy of P. We show how to establish this for various classes of maps. For planar P, the existence of limn → ∞ | M n| 1 n suffices. For nonplanar P, we require more detailed asymptotic information.

2-connected coverings of bounded degree in 3-connected graphs

In a recent paper, Barnette showed that every 3-connected planar graph has a 2-connected spanning subgraph of maximum degree at most fifteen, he also constructed a planar triangulation that does not have 2-connected spanning subgraphs of maximum degree five. In this paper, we show that every 3-connected graph which is embeddable in the sphere, the projective plane, the torus or the Klein bottle has a 2-connected spanning subgraph of maximum degree at most six.

Diagonal Flips in Labelled Planar Triangulations

Abstract. A classical result of Wagner states that any two (unlabelled) planar triangulations with the same number of vertices can be transformed into each other by a finite sequence of diagonal flips. Recently Komuro gives a linear bound on the maximum number of diagonal flips needed for such a transformation. In this paper we show that any two labelled triangulations can be transformed into each other using at most O(nlogn) diagonal flips. We will also show that any planar triangulation with n>4 vertices has at least n− 2 flippable edges. Finally, we prove that if the minimum degree of a triangulation is at least 4, then it contains at least 2n + 3 flippable edges. These bounds can be achieved by an infinite class of triangulations.

SHARP CONCENTRATION OF THE NUMBER OF SUBMAPS IN RANDOM PLANAR TRIANGULATIONS

We show that the maximum vertex degree in a random 3-connected planar triangulation is concentrated in an interval of almost constant width. This is a slightly weaker type of result than our earlier determination of the limiting distribution of the maximum vertex degree in random planar maps and in random triangulations of a (convex) polygon. We also derive sharp concentration results on the number of vertices of given degree in random planar maps of all three types. Some sharp concentration results about general submaps in 3-connected triangulations are also given.

Convex Programming and Circumference of 3-Connected Graphs of Low Genus

The circumference of a graphGis the length of a longest cycle inG. In this paper, we shall show that, ifGis a 3-connected graph embeddable in the plane, the projective plane, the torus, or the Klein bottle, thenGhas circumference at least (1/6)×|V(G)|0.4+1. This improves a result of Jackson and Wormald.