Abstract
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 is an element of E(G) and |c(u)-c(v)| }= 2 if uv is an element of E(T). Broersma et al. [J.
Graph Theory, 55 (2007), pp. 137-152] asked whether there exists a constant c such that for every triangle-free graph G with an arbitrary spanning tree T the inequality BBC(G, T) {= chi(G) c holds. We answer this question negatively by showing the existence of triangle-free graphs R_n and their spanning trees T_n such that BBC(R-n, T-n) = 2 chi(R-n)-1 = 2n-1.
In order to answer the question, we obtain a result of independent interest. We modify the well-known Mycielski construction and construct triangle-free graphs J(n) for every integer n, with chromatic number n and 2-tuple chromatic number 2n (here 2 can be replaced by any integer t).