3-trees with few vertices of degree 3 in circuit graphs

A circuit graph (G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most n-7/3 vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F(chi) with Euler characteristic chi >= 0 has a 3-tree with at most n/3 + c(chi) vertices of degree 3, where c(chi) is a constant depending only on F(chi). (C) 2008 Elsevier B.V. All rights reserved.