Triangle-free graphs of tree-width t are inverted right perpendicular (t+3)/2 inverted left perpendicular-colorable

We prove that every triangle-free graph of tree-width t has chromatic number at most inverted right perpendicular (t + 3)/2 inverted left perpendicular, and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t. (C) 2017 Elsevier Ltd. All rights reserved.