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 vertical bar c(u)-c(v)vertical bar >= 1 if uv is an element of E(G) and vertical bar c(u)-c(v)vertical bar >= 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).
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Univ Pavol Jozef Safarik, Fac Sci, Inst Math, Kosice 04154, SlovakiaUniv Pavol Jozef Safarik, Fac Sci, Inst Math, Kosice 04154, Slovakia
Miskuf, Jozef
Skrekovski, Riste
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Univ Ljubljana, Dept Math, Fac Math & Phys, Ljubljana 1000, SloveniaUniv Pavol Jozef Safarik, Fac Sci, Inst Math, Kosice 04154, Slovakia
Skrekovski, Riste
Tancer, Martin
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Charles Univ Prague, Dept Appl Math, Fac Math & Phys, CR-11800 Prague, Czech Republic
Charles Univ Prague, Fac Math & Phys, Inst Theoret Comp Sci, Prague 11800, Czech RepublicUniv Pavol Jozef Safarik, Fac Sci, Inst Math, Kosice 04154, Slovakia
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Inst Teknol Bandung, Fac Math & Nat Sci, Combinatorial Math Res Div, J1 Ganesa 10, Bandung 40132, IndonesiaInst Teknol Bandung, Fac Math & Nat Sci, Combinatorial Math Res Div, J1 Ganesa 10, Bandung 40132, Indonesia
Saputro, S. W.
Salman, A. N. M.
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Inst Teknol Bandung, Fac Math & Nat Sci, Combinatorial Math Res Div, J1 Ganesa 10, Bandung 40132, IndonesiaInst Teknol Bandung, Fac Math & Nat Sci, Combinatorial Math Res Div, J1 Ganesa 10, Bandung 40132, Indonesia