15,153 research outputs found
Graph Colouring with Input Restrictions
In this thesis, we research the computational complexity of the graph colouring problem and its variants including precolouring extension and list colouring for graph classes that can be characterised by forbidding one or more induced subgraphs. We investigate the structural properties of such graph classes and prove a number of new properties. We then consider to what extent these properties can be used for efficiently solving the three types of colouring problems on these graph classes. In some cases we obtain polynomial-time algorithms, whereas other cases turn out to be NP-complete.
We determine the computational complexity of k-COLOURING, k-PRECOLOURING EXTENSION and LIST k-COLOURING on -free graphs. In particular, we prove that k-COLOURING on -free graphs is NP-complete, 4-PRECOLOURING EXTENSION -free graphs is NP-complete, and LIST 4-COLOURING on -free graphs is NP-complete. In addition, we show the existence of an integer r such that k-COLOURING is NP-complete for -free graphs with girth 4. In contrast, we determine for any fixed girth a lower bound such that every -free graph with girth at least is 3-colourable. We also prove that 3-LIST COLOURING is NP-complete for complete graphs minus a matching. We present a polynomial-time algorithm for solving 4-PRECOLOURING EXTENSION on -free graphs, a polynomial-time algorithm for solving LIST 3-Colouring on -free graphs, and a polynomial-time algorithm for solving LIST 3-COLOURING on -free graphs. We prove that LIST k-COLOURING for -free graphs is also polynomial-time solvable. We obtain several new dichotomies by combining the above results with some known results
Colouring (P_r+P_s)-Free Graphs
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs
Colouring (Pr + Ps)-Free Graphs
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u)⊆{1,…,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph Pr+Ps is the disjoint union of the r-vertex path Pr and the s-vertex path Ps. We prove that List 3-Colouring is polynomial-time solvable for (P2+P5)-free graphs and for (P3+P4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices
Colouring (Pr+Ps)-free graphs.
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs
Clustered Colouring in Minor-Closed Classes
The "clustered chromatic number" of a class of graphs is the minimum integer
such that for some integer every graph in the class is -colourable
with monochromatic components of size at most . We prove that for every
graph , the clustered chromatic number of the class of -minor-free graphs
is tied to the tree-depth of . In particular, if is connected with
tree-depth then every -minor-free graph is -colourable with
monochromatic components of size at most . This provides the first
evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about
defective colouring of -minor-free graphs. If then we prove that 4
colours suffice, which is best possible. We also determine those minor-closed
graph classes with clustered chromatic number 2. Finally, we develop a
conjecture for the clustered chromatic number of an arbitrary minor-closed
class
- …