Triangle-free graphs of tree-width t are ⌈(t+3)∕2⌉-colorable

We prove that every triangle-free graph of tree-width t has chromatic number at most ⌈(t+3)∕2⌉, 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.