Nonrepetitive colorings of trees

Brešar, Boštjan; Grytczuk, Jarosław; Klavar, Sandi; Niwczyk, Stanisław; Peterin, Iztok

doi:10.1016/j.disc.2006.06.017

Cited by 40 publications

(42 citation statements)

References 8 publications

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“…A nonexhaustive list of these papers contains [3,9,4,1,8,6] and [7]. In [2] some properties related to the palindrome complexity of finite words are considered.…”

Section: Introduction and Notationsmentioning

confidence: 99%

Total palindrome complexity of finite words

Anisiu

Kása

2010

Discrete Mathematics

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a b s t r a c tThe palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).

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“…A nonexhaustive list of these papers contains [3,9,4,1,8,6] and [7]. In [2] some properties related to the palindrome complexity of finite words are considered.…”

Section: Introduction and Notationsmentioning

confidence: 99%

Total palindrome complexity of finite words

Anisiu

Kása

2010

Discrete Mathematics

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“…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].…”

Section: Nonrepetitive Graph Colouringsmentioning

confidence: 99%

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

Dujmović

Morin

Wood

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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Abstract-Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed families. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. We use layered separators to prove O(log n) bounds for a number of problems where O( √ n) was a long standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results to all proper minor-closed families, with a O(log n) bound on the nonrepetitive chromatic number, and a log O(1) n bound on the queue-number. Only for planar graphs were log O(1) n bounds previously known. Our results imply that every graph from a proper minor-closed class has a 3-dimensional grid drawing with n log O(1) n volume, whereas the previous best bound was O(n 3/2 ). Readers interested in the full details should consult arXiv:1302.0304 and arXiv:1306.1595, rather than the current extended abstract.

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“…The Thue chromatic number of graphs has been studied in many papers [2,4,6,11,8]. Upper and lower bounds for the Thue chromatic number of many classes of graphs are obtained in the literature.…”

Section: Introductionmentioning

confidence: 99%

Fractional Thue chromatic number of graphs

Zhong

Zhu

2016

Discrete Applied Mathematics

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a b s t r a c tThis paper introduces the concept of the fractional Thue chromatic number of graphs and studies the relation between the fractional Thue chromatic number and the Thue chromatic number. We determine the fractional Thue chromatic number of all paths, all trees with no vertices of degree two, and all cycles, except C 10 , C 14 , C 17 . As a consequence, we prove that if G is a path or a tree with no degree two vertices, then its fractional Thue chromatic number equals its Thue chromatic number. On the other hand, we show that there are trees and cycles whose fractional Thue chromatic numbers are strictly less than their Thue chromatic numbers.

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Nonrepetitive colorings of trees

Cited by 40 publications

References 8 publications

Total palindrome complexity of finite words

Total palindrome complexity of finite words

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

Fractional Thue chromatic number of graphs

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