Fundamental cycles and graph embeddings

Abstract

In 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 C ₁,C ₂,…,C _2g with C _2i−1 ∩ C _2i ≠ /0 for 1 ⩽ i ⩽ g. In particular, among β(G) fundamental cycles of any spanning tree T of a graph G, there are exactly 2_γM (G) cycles C ₁, C ₂,…,C _2γM(G) such that C _2i−1 ∩ C _2i ≠ /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 G E(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 $O((\beta (G))^{\frac{5} {2}} )$ 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.