6 research outputs found
Anagram-free Graph Colouring
An anagram is a word of the form where is a non-empty word and
is a permutation of . We study anagram-free graph colouring and give bounds
on the chromatic number. Alon et al. (2002) asked whether anagram-free
chromatic number is bounded by a function of the maximum degree. We answer this
question in the negative by constructing graphs with maximum degree 3 and
unbounded anagram-free chromatic number. We also prove upper and lower bounds
on the anagram-free chromatic number of trees in terms of their radius and
pathwidth. Finally, we explore extensions to edge colouring and
-anagram-free colouring.Comment: Version 2: Changed 'abelian square' to 'anagram' for consistency with
'Anagram-free colourings of graphs' by Kam\v{c}ev, {\L}uczak, and Sudakov.
Minor changes based on referee feedbac
Anagram-Free Chromatic Number is not Pathwidth-Bounded
The anagram-free chromatic number is a new graph parameter introduced
independently Kam\v{c}ev, {\L}uczak, and Sudakov (2017) and Wilson and Wood
(2017). In this note, we show that there are planar graphs of pathwidth 3 with
arbitrarily large anagram-free chromatic number. More specifically, we describe
-vertex planar graphs of pathwidth 3 with anagram-free chromatic number
. We also describe vertex graphs with pathwidth
having anagram-free chromatic number in .Comment: 8 pages, 3 figure
Avoiding abelian powers cyclically
We study a new notion of cyclic avoidance of abelian powers. A finite word avoids abelian -powers cyclically if for each abelian -power of period occurring in the infinite word , we have . Let be the least integer such that for all there exists a word of length over a -letter alphabet that avoids abelian -powers cyclically. Let be the least integer such that there exist arbitrarily long words over a -letter alphabet that avoid abelian -powers cyclically.We prove that , , , and for . Moreover, we show that , , and .</p