Backbone coloring for triangle-free planar graphs

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G, H) is a mapping f: V(G) → {1, 2, ···, k} such that |f(u) − f(v)| ≥ 2 if uv ∈ E(H) and |f(u) − f(v)| ≥ 1 if uv ∈ E(G)E(H). The backbone chromatic number of (G, H) denoted by χb(G, H) is the smallest integer k such that (G, H) has a backbone-k-coloring. In this paper, we prove that if G is either a connected triangle-free planar graph or a connected graph with mad(G) < 3, then there exists a spanning tree T of G such that χb(G, T) ≤ 4.