Nonrepetitive colorings of graphs of bounded tree-width

Abstract

A sequence of the form s 1 s 2 . . . s m s 1 s 2 . . . s m is called a repetition. A vertex-coloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [Thue type problems for graphs, points and numbers, manuscript] by proving that every outerplanar graph has a nonrepetitive 12-coloring. We also show that graphs of tree-width t have nonrepetitive 4 t -colorings.

Layered Separators for Queue Layouts, 3D Graph Drawing and Nonrepetitive Coloring

“…The following lemma generalises a result by Kündgen and Pelsmajer [17], who proved it when each bag of the tree decomposition is a clique (that is, for chordal graphs). We allow bags to induce more general graphs.…”

“…A number of graph classes are known to have bounded nonrepetitive chromatic number. In particular, trees are nonrepetitively 4-colourable [16], [17], outerplanar graphs are nonrepetitively 12-colourable [17], [18], and more generally, every graph with treewidth k is nonrepetitively 4 k -colourable [17]. Graphs with maximum degree Δ are nonrepetitively O(Δ 2 )-colourable [19].…”

“…However, in this section we extend our methods from previous sections to prove that graphs from any proper minor-closed class have nonrepetitive chromatic number O(log n), track/queue-number log O(1) n, and 3-dimensional grid drawings with n log O(1) n volume. [17] and implicitly by Dujmović etal [12]. It is a key to the proof that graphs of bounded treewidth have bounded nonrepetitive chromatic number [17] and bounded track-number [12].…”

“…[17] and implicitly by Dujmović etal [12]. It is a key to the proof that graphs of bounded treewidth have bounded nonrepetitive chromatic number [17] and bounded track-number [12].…”

Nonrepetitive colorings of graphs

“…Kündgen and Pelsmajer [4] and Barát and Varjú [3] proved independently that π 2 (G) is bounded for graphs of bounded treewidth. By the result of Robertson and Seymour [5] it follows that if H is any fixed planar graph then π k (G) is bounded for graphs not containing H as a minor.…”