On 2-Connected Spanning Subgraphs with Low Maximum Degree

Given a graphG, let ak-trestle ofGbe a 2-connected spanning subgraph ofGof maximum degree at mostk. Also, letχ(G) be the Euler characteristic ofG. This paper shows that every 3-connected graphGhas a (10−2χ(G))-trestle. Ifχ(G)⩽−5, this is improved to 8−2χ(G), and forχ(G)⩽−10, this is further improved to 6−2χ(G). This result is shown to be best possible for almost all values ofχ(G) by the demonstration of 3-connected graphs embedded on each surface of Euler characteristicχ⩽0 which have no (5−2χ)-trestle. Also, it is shown that a 4-connected graph embeddable on a surface with non-negative Euler characteristic has a 3-trestle, approaching a conjecture of Nash-Williams.