Steinberg-like theorems for backbone colouring

Inspired by Steinberg's conjecture, we conjecture that if G is a planar graph containing no cycles on 4 or 5 vertices and H⊆G is a forest, then CBC2(G,H)≤6. In this work, we first show that if G is a planar graph containing no cycle on 4 or 5 vertices and H⊆G is a forest, then CBC2(G,H)≤7. Then, we prove that if H⊆G is a forest whose connected components are paths, then CBC2(G,H)≤6.