22 research outputs found
Restricted coloring problems on graphs with few P4's
International audienceIn this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P4-tidy graphs and (q,q − 4)-graphs, for every fixed q. These classes include cographs, P4-sparse and P4-lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of (q,q − 4)-graphs. All these coloring problems are known to be NP-hard for general graphs
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM
Vertex colouring and forbidden subgraphs - a survey
There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions
Relative Timing Information and Orthology in Evolutionary Scenarios
Evolutionary scenarios describing the evolution of a family of genes within a
collection of species comprise the mapping of the vertices of a gene tree
to vertices and edges of a species tree . The relative timing of the last
common ancestors of two extant genes (leaves of ) and the last common
ancestors of the two species (leaves of ) in which they reside is indicative
of horizontal gene transfers (HGT) and ancient duplications. Orthologous gene
pairs, on the other hand, require that their last common ancestors coincides
with a corresponding speciation event. The relative timing information of gene
and species divergences is captured by three colored graphs that have the
extant genes as vertices and the species in which the genes are found as vertex
colors: the equal-divergence-time (EDT) graph, the later-divergence-time (LDT)
graph and the prior-divergence-time (PDT) graph, which together form an edge
partition of the complete graph.
Here we give a complete characterization in terms of informative and
forbidden triples that can be read off the three graphs and provide a
polynomial time algorithm for constructing an evolutionary scenario that
explains the graphs, provided such a scenario exists. We show that every EDT
graph is perfect. While the information about LDT and PDT graphs is necessary
to recognize EDT graphs in polynomial-time for general scenarios, this extra
information can be dropped in the HGT-free case. However, recognition of EDT
graphs without knowledge of putative LDT and PDT graphs is NP-complete for
general scenarios. In contrast, PDT graphs can be recognized in
polynomial-time. We finally connect the EDT graph to the alternative
definitions of orthology that have been proposed for scenarios with horizontal
gene transfer. With one exception, the corresponding graphs are shown to be
colored cographs
Topics in graph colouring and extremal graph theory
In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let be a connected graph with vertices and maximum degree . Let denote the graph with vertex set all proper -colourings of and two -colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that has a unique non-trivial component with diameter . This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours , of a component of the subgraph induced by vertices with colour or . Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all -colourings of a graph are Kempe equivalent unless is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees