Graphs and Combinatorics (1994) 10:123-131 Graphs and Combinatorics 9 Springer-Verlag 1994 Spanning Eulerian Subgraphs of Bounded Degree in Triangulations Zhicheng Gao I and Nicholas C. Wormald 2 Dept of Mathematics and Statistics Carleton University Ottawa Canada K1S 5B6 2 Department of Mathematics University of Melbourne Parkville, VIC 3052, Australia Abstract. We show that every triangulation of a disk or an annulus has a spanning Eulerian subgraph with maximum degree eight. Since every triangulation in the projective plane, the torus and the Klein bottle has a spanning subgraph which triangulates an annulus, this implies that all triangulations in the projective plane, the torus and the Klein bottle have spanning Eulerian subgraphs with maximum degree at most eight. 1. Introduction Throughout the paper, all graphs are finite and have no loops or multiple edges. If a graph G is embedded in a surface Z, the representativity of G is the smallest number of intersections a noncontractible simple closed curve in s has with G. (If s is the sphere, the representativity may be defined as oo.) Let cg be a collection of cycles in G, an embedding of G in the plane is called a cg-triangulation if every cycle in cg bounds a face and all other faces are triangles. A {C}-triangulation is simply denoted by C-triangulation and is also called a triangulation of a disk; A {C1, C2 }-triangula- tion is also called a triangulation of an annulus. A (closed) k-walk of a graph is a (closed) walk which visits every vertex at least once and at most k times. A k-walk is called a k-trail if it has no repeated edges. Clearly in a graph with more than two vertices a 1-walk (closed 1-walk) is just a Hamilton path (cycle), and a closed k-trail for any k is a spanning Eulerian subgraph with maximum degree at most 2k. There is a close relationship between k-walks and spanning trees. Jackson and Wormald [7] proved that if a graph has a closed k-walk, then it has a spanning tree with maximum degree at most k + 1, and observed on the other hand that if a graph has a spanning tree with maximum degree at most k then it has a closed k-walk. The study of spanning subgraphs with certain properties is one of the most important area in graph theory. Whitney started a chain of papers which successively proved that: * Research supported by the Australian Research Council