6 research outputs found
On vertex coloring without monochromatic triangles
We study a certain relaxation of the classic vertex coloring problem, namely,
a coloring of vertices of undirected, simple graphs, such that there are no
monochromatic triangles. We give the first classification of the problem in
terms of classic and parametrized algorithms. Several computational complexity
results are also presented, which improve on the previous results found in the
literature. We propose the new structural parameter for undirected, simple
graphs -- the triangle-free chromatic number . We bound by
other known structural parameters. We also present two classes of graphs with
interesting coloring properties, that play pivotal role in proving useful
observation about our problem. We give/ask several conjectures/questions
throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac
Graph Partitioning With Input Restrictions
In this thesis we study the computational complexity of a number of graph
partitioning problems under a variety of input restrictions. Predominantly,
we research problems related to Colouring in the case where the input
is limited to hereditary graph classes, graphs of bounded diameter or some
combination of the two.
In Chapter 2 we demonstrate the dramatic eect that restricting our
input to hereditary graph classes can have on the complexity of a decision
problem. To do this, we show extreme jumps in the complexity of three
problems related to graph colouring between the class of all graphs and every
other hereditary graph class.
We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be
H-free for some graph H if it contains no induced subgraph isomorphic to
H. Similarly, G is said to be H-free for some set of graphs H, if it does not
contain any graph in H as an induced subgraph. Here, the set H consists
usually of a single cycle or tree but may also contain a number of cycles, for
example we give results for graphs of bounded diameter and girth.
Chapter 4 is dedicated to three variants of the Colouring problem,
Acyclic Colouring, Star Colouring, and Injective Colouring.
We give complete or almost complete dichotomies for each of these decision
problems restricted to H-free graphs.
In Chapter 5 we study these problems, along with three further variants of
3-Colouring, Independent Odd Cycle Transversal, Independent
Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of
bounded diameter.
Finally, Chapter 6 deals with a dierent variety of problems. We study
the problems Disjoint Paths and Disjoint Connected Subgraphs for
H-free graphs