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

A circuit graph  (G,C)(G,C) is a 2-connected plane graph GG with an outer cycle CC such that from each inner vertex vv, there are three disjoint paths to CC. In this paper, we shall show that a circuit graph with nn vertices has a 3-tree   (i.e., a spanning tree with maximum degree at most 3) with at most n−73 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 nn vertices on a surface FχFχ with Euler characteristic χ≥0χ≥0 has a 3-tree with at most n3+cχ vertices of degree 3, where cχcχ is a constant depending only on FχFχ.