Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-11T03:24:29.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Anagram-Free Colourings of Graphs

Part of: Graph theory

Published online by Cambridge University Press:  08 August 2017

NINA KAMČEV ,
TOMASZ ŁUCZAK  and
BENNY SUDAKOV
NINA KAMČEV
Affiliation:
Department of Mathematics, ETH, Rämistrasse 101, 8092 Zurich, Switzerland (e-mail: nina.kamcev@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
TOMASZ ŁUCZAK
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland (e-mail: tomasz@amu.edu.pl)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, Rämistrasse 101, 8092 Zurich, Switzerland (e-mail: nina.kamcev@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
Get access Rights & Permissions [Opens in a new window]

Abstract

A sequence S is called anagram-free if it contains no consecutive symbols r1r2. . .rkrk+1. . .r2k such that rk+1. . .r2k is a permutation of the block r1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graph G is called anagram-free if the sequence of colours on any path in G is anagram-free. We call the minimal number of colours needed for such a colouring the anagram-chromatic number of G.

In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we essentially give each vertex a separate colour.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Special Issue 4: Special Issue: Oberwolfach Workshop , July 2018 , pp. 623 - 642
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alon, N. and Friedland, S. (2008) The maximum number of perfect matchings in graphs with a given degree sequence. Electron. J. Combin. 15 (1) note 13.CrossRefGoogle Scholar
[2] Alon, N., Grytczuk, J., Hałuszczak, M. and Riordan, O. (2002) Non-repetitive colorings of graphs. Random Struct. Alg. 21 336–346.CrossRefGoogle Scholar
[3] Alon, N., Grytczuk, J., Lasoń, M. and Michałek, M. (2009) Splitting necklaces and measurable colorings of the real line. Proc. Amer. Math. Soc. 137 1593–1599.CrossRefGoogle Scholar
[4] Alon, N., Seymour, P. and Thomas, R. (1990) A separator theorem for graphs with an excluded minor and its applications. In STOC '90: Proc. 22nd Annual ACM Symposium on Theory of Computing, ACM, pp. 293–299.CrossRefGoogle Scholar
[5] Bean, D. R., Ehrenfeucht, A. and McNulty, G. F. (1979) Avoidable patterns in strings of symbols. Pacific J. Math. 85 261–294.CrossRefGoogle Scholar
[6] Brown, T. C. (1971) Is there a sequence on four symbols in which no two adjacent segments are permutations of one another? Amer. Math. Monthly 78 886–888.CrossRefGoogle Scholar
[7] Cheilaris, P., Keszegh, B. and Pálvölgyi, D. (2013) Unique-maximum and conflict-free coloring for hypergraphs and tree graphs. SIAM J. Discrete Math 27 1775–1787.CrossRefGoogle Scholar
[8] Chen, C. C. and Quimpo, N. (1981) On strongly Hamiltonian abelian group graphs. In Combinatorial Mathematics VIII (McAvaney, K. L., ed.), Vol. 884 of Lecture Notes in Mathematics, Springer, pp. 23–34.CrossRefGoogle Scholar
[9] Dekking, F. M. (1979) Strongly non-repetitive sequences and progression free sets. J. Combin. Theory Ser. A 27 181–185.CrossRefGoogle Scholar
[10] Erdős, P. (1961) Some unsolved problems. Magyar Tud. Akad. Mat. Kutato. Int. Kozl. 6 221–254.Google Scholar
[11] Evdokimov, A. A. (1968) Strongly asymmetric sequences generated by finite number of symbols. Dokl. Akad. Nauk. SSSR 179 1268–1271. Soviet Math. Dokl. 9 536–539.Google Scholar
[12] Frieze, A. M. and Łuczak, T. (1992) On the independence and chromatic numbers of random regular graphs. J. Combin. Theory Ser. B 54 123–132.CrossRefGoogle Scholar
[13] Grytczuk, J. (2007) Nonrepetitive colorings of graphs: A survey. Int. J. Math. Math. Sci. 2007 74639.CrossRefGoogle Scholar
[14] Grytczuk, J. (2008) Thue type problems for graphs, points, and numbers. Discrete Math. 308 4419–4429.CrossRefGoogle Scholar
[15] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.CrossRefGoogle Scholar
[16] Keränen, V. (1992) Abelian squares are avoidable on 4 letters. In ICALP 1992: International Colloquium on Automata, Languages, and Programming, Vol. 623 of Lecture Notes in Computer science, Springer, pp. 41–52.Google Scholar
[17] Krivelevich, M., Lubetzky, E. and Sudakov, B. (2010) Hamiltonicity thresholds in Achlioptas processes. Random Struct. Alg. 37 1–24.CrossRefGoogle Scholar
[18] Novikov, P. S. and Adjan, S. I. (1968) On infinite periodic groups I, II, III. Math. USSR Izv. 32 212–244, 251–524, 709–731.Google Scholar
[19] Pleasants, P. A. B. (1970) Non-repetitive sequences. Proc. Cambridge Philos. Soc. 68 267–274.CrossRefGoogle Scholar
[20] Thue, A. (1906) Über unendliche Zeichenreichen. Norske Videnskabers Selskabs Skrifter, I Mathematisch-Naturwissenschaftliche Klasse, Christiania 7 1–22.Google Scholar
[21] Wilson, T. E. and Wood, D. R. (2016) Abelian square-free graph colouring. arXiv1607.01117Google Scholar