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 frac(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_χ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + c_χ vertices of degree 3, where c_χ is a constant depending only on F_χ.