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 If（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 Xb（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 Xb（G, T） ≤ 4.