Backbone Colorings and Generalized Mycielski’s Graph

For a graph G and its spanning tree T the backbone chromatic number, BBC(G,T ), is defined as the minimum k such that there exists a coloring c : V (G) → {1, 2, . . . , k} satisfying |c(u) − c(v)| ≥ 1 if uv ∈ E(G) and |c(u)− c(v)| ≥ 2 if uv ∈ E(T ). Broersma et al. [1] asked whether there exists a constant c such that for every triangle-free graphG with an arbitrary spanning tree T the inequality BBC(G,T ) ≤ χ(G) + c holds. We answer this question negatively by showing the existence of triangle-free graphs Rn and their spanning trees Tn such that BBC(Rn, Tn) = 2χ(Rn)− 1 = 2n− 1. In order to answer the question we obtain a result of independent interest. We modify the well known Mycielski’s construction and construct triangle-free graphs Jn, for every integer n, with chromatic number n and 2-tuple chromatic number 2n (here 2 can be replaced by any integer t).