Pascal Ochem

Strong edge-colouring and induced matchings

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of three edges uses three colours. An induced matching of a graph G is a subset I of edges of G such that the graph induced by the endpoints of I is a matching. In this paper, we prove the NP-completeness of strong 4-, 5-, and 6-edge-colouring and maximum induced matching in some subclasses of subcubic triangle-free planar graphs. We also obtain a tight upper bound for the minimum number of colours in a strong edge-colouring of outerplanar graphs as a function of the maximum degree.

Vertex Partitions of Graphs into Cographs and Stars

A cograph is a graph that contains no path on four vertices as an induced subgraph. A cograph k-partition of a graph G = (V, E) is a vertex-partition of G into k sets V1 , . . . , Vk ⊂ V so that the graph induced by Vi is a cograph for 1 ≤ i ≤ k. Gimbel and Nesetril [5] studied the complexity aspects of the cograph k-partitions and raised the following questions: Does there exist a triangle-free planar graph that is not cograph 2-partitionable? If the answer is yes, what is the complexity of the associated decision problem? In this paper, we prove that such an example exists and that deciding whether a triangle-free planar graph admits a cograph 2-partition is NP-complete. We also show that every graph with maximum average degree at most 14/5 admits a cograph 2-partition such that each component is a star on at most three vertices.

BINARY WORDS AVOIDING THE PATTERN AABBCABBA

We show that there are three types of infinite words over the two-letter alphabet {0,1} that avoid the pattern AABBCABBA. These types, P, E 0 , and E 1 , differ by the factor complexity and the asymptotic frequency of the letter 0. Type P has polynomial factor complexity and letter frequency ½ . Type E o has exponential factor complexity and the frequency of the letter 0 is at least 0.45622 and at most 0.48684. Type E 1 is obtained from type E 0 by exchanging 0 and 1.

Binary patterns in binary cube-free words: Avoidability and growth

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.

The complexity of partitioning into disjoint cliques and a triangle-free graph

Motivated by Chudnovskys structure theorem of bull-free graphs, Abu-Khzam, Feghali, and Muller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs.

Sieve methods for odd perfect numbers

Using a new factor chain argument, we show that 5 does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10^8 < p < 10^1000. These results are generalized to much broader situations.

Oriented coloring of triangle-free planar graphs and 2-outerplanar graphs

A graph is planar if it can be embedded on the plane without edge-crossings. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239–249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215–219, 2007].

A short proof that shuffle squares are 7-avoidable

A shuffle square is a word that can be partitioned into two identical words. We obtain a short proof that there exist exponentially many words over the 7 letter alphabet containing no shuffle square as a factor. The method is a generalization of the so-called power series method using ideas of the entropy compression method as developped by Goncalves et al. [Entropy compression method applied to graph colorings.

Characterization of some binary words with few squares

Thue proved that the factors occurring infinitely many times in square-free words over { 0 , 1 , 2 } avoiding the factors in { 010 , 212 } are the factors of the fixed point of the morphism 0 ? 012 , 1 ? 02 , 2 ? 1 . He similarly characterized square-free words avoiding { 010 , 020 } and { 121 , 212 } as the factors of two morphic words. In this paper, we exhibit smaller morphisms to define these two square-free morphic words and we give such characterizations for six types of binary words containing few distinct squares.

Avoiding or Limiting Regularities in Words

It is commonly admitted that the origin of combinatorics on words goes back to the work of Axel Thue in the beginning of the twentieth century, with his results on repetition-free words. Thue showed that one can avoid cubes on infinite binary words and squares on ternary words. Up to now, a large part of the work on the theoretic part of combinatorics on words can be viewed as extensions or variations of Thue’s work, that is, showing the existence (or nonexistence) of infinite words avoiding, or limiting, a repetition-like pattern. The goal of this chapter is to present the state of the art in the domain and also to present general techniques used to prove a positive or a negative result. Given a repetition pattern P and an alphabet, we want to know if an infinite word without P exists. If it exists, we are also interested in the size of the language of words avoiding P, that is, the growth rate of the language. Otherwise, we are interested in the minimum number of factors P that a word must contain. We talk about limitation of usual, fractional, abelian, and k-abelian repetitions and other generalizations such as patterns and formulas. The last sections are dedicated to the presentation of general techniques to prove the existence or the nonexistence of an infinite word with a given property.