On square-free vertex colorings of graphs

Abstract

A sequence of symbols a1, a2 . . . is called square-free if it does not contain a subsequence of consecutive terms of the form x1, . . . , xm, x1, . . . , xm. A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Ha luszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k-tree have a coloring of this kind using O(c k ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

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

“…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].…”

Nonrepetitive list colourings of paths

“…This result was extended in [5] and [18] for general k-trees so that π(G) ≤ 4 k for every graph G of treewidth at most k. Hence, by the famous result of Robertson and Seymour [25] (see [11] for an excellent exposition), for every fixed planar graph H, the class of graphs not containing H as a minor has bounded Thue chromatic number. Whether π(G) is bounded for the class of planar graphs remains a challenging open problem of the area.…”

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.…”