Han Ren

Generating cycle spaces for graphs on surfaces with small genera

In this paper we investigate cycle base structures of a 2-(edge)-connected graph on surfaces with small genera and show that short cycles do generate the cycle spaces in the case of small face-embeddings. Examples show that the conditions in our main results are best possible. As applications we find the exact formulae for the minimum lengths of cycle bases of some types of graphs which conclude several known results. Finally, we generalize those results for graphs embedded on general orientable surfaces.

The Number of Loopless 4-Regular Maps on the Projective Plane

In this paper rooted loopless (near) 4-regular maps on surfaces such as the sphere and the projective plane are counted and exact formulae with up to three or four parameters for such maps are given. Several classical results on regular maps and one-faced maps are deduced.

4-Regular Maps on the Klein Bottle

In this paper the number of rooted near-4-regular maps on the Klein bottle is investigated and formulae with up to four parameters are given. Also the number of rooted near-4-regular maps with exactly k nonroot-vertex loops on this surface is provided. In particular, explicit formulae for counting (loopless) 4-regular maps on this surface are deduced.

Short cycle structures for graphs on surfaces and an open problem of Mohar and Thomassen

In this paper we investigate cycle base structures of a (weighted) graph and show that much information of short cycles is contained in an MCB (i.e., minimum cycle base). After setting up a Hall type theorem for base-transformation, we give a sufficient and necessary condition for a cycle base to be an MCB. Furthermore, we show that the structure of MCB in a (weighted) graph is unique. The property is also true for those having a longest length (although much work has been down in evaluating MCB, little is known for those having a longest length). We use those methods to find out some unknown properties for short cycles sharing particular properties in (unweighted) graphs. As applications, we determine the structures of short cycles in an embedded graph and show that there exist polynomially bounded algorithms in finding a shortest contractible cycle and a shortest two-sided cycle provided such cycles exist. Those answer an open problem of B. Mohar and C. Thomassen.

Enumeration of 2 -connected Loopless 4 -regular Maps on the Plane

In this paper rooted (near-) 4-regular maps on the plane are counted with respect to the root-valency, the number of edges, the number of inner faces, the number of non-root vertex loops, the number of non-root vertex blocks, and the number of multi-edges. As special cases, formulae of several types of rooted 4-regular maps such as 2-connected 4-regular planar maps, rooted 2-connected (connected) 4-regular planar maps without loops are also presented. Several known results on 4-regular maps and trees of Tutte are also concluded. Finally, asymptotic formulae for the numbers of those types of maps are given.

Fundamental cycles and graph embeddings

AbstractIn this paper, we investigate fundamental cycles in a graph G and their relations with graph embeddings. We show that a graph G may be embedded in an orientable surface with genus at least g if and only if for any spanning tree T, there exists a sequence of fundamental cycles C1,C2,…,C2g with C2i−1 ∩ C2i ≠ /0 for 1 ⩽ i ⩽ g. In particular, among β(G) fundamental cycles of any spanning tree T of a graph G, there are exactly 2γM(G) cycles C1, C2,…,C2γM(G) such that C2i−1 ∩ C2i ≠ /0 for 1 ⩽ i ⩽ γM(G), where β(G) and γM(G) are the Betti number and the maximum genus of G, respectively. This implies that it is possible to construct an orientable embedding with large genus of a graph G from an arbitrary spanning tree T (which may have very large number of odd components in GE(T)). This is different from the earlier work of Xuong and Liu, where spanning trees with small odd components are needed. In fact, this makes a common generalization of Xuong, Liu and Fu et al. Furthermore, we show that (1) this result is useful for locating the maximum genus of a graph having a specific edge-cut. Some known results for embedded graphs are also concluded; (2) the maximum genus problem may be reduced to the maximum matching problem. Based on this result and the algorithm of Micali-Vazirani, we present a new efficient algorithm to determine the maximum genus of a graph in

BISINGULAR MAPS ON SOME SURFACES

nO((beta (G))^{frac{5}n{2}} )n

Minor and minimum cycle bases of a 3-connected planar graph

steps. Our method is straight and quite different from the algorithm of Furst, Gross and McGeoch which depends on a result of Giles where matroid parity method is needed.