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Proper connection number of graphs
The concept of \emph{proper connection number} of graphs is an extension of proper colouring and is motivated by rainbow connection number of graphs. Let be an edge-coloured graph. Andrews et al.\cite{Andrews2016} and, independently, Borozan et al.\cite{Borozan2012} introduced the concept of proper connection number as follows: A coloured path in an edge-coloured graph is called a \emph{properly coloured path} or more simple \emph{proper path} if two any consecutive edges receive different colours. An edge-coloured graph is called a \emph{properly connected graph} if every pair of vertices is connected by a proper path. The \emph{proper connection number}, denoted by , of a connected graph is the smallest number of colours that are needed in order to make properly connected. Let be an integer. If every two vertices of an edge-coloured graph are connected by at least proper paths, then is said to be a \emph{properly -connected graph}. The \emph{proper -connection number} , introduced by Borozan et al. \cite{Borozan2012}, is the smallest number of colours that are needed in order to make a properly -connected graph.
The aims of this dissertation are to study the proper connection number and the proper 2-connection number of several classes of connected graphs. All the main results are contained in Chapter 4, Chapter 5 and Chapter 6.
Since every 2-connected graph has proper connection number at most 3 by Borozan et al. \cite{Borozan2012} and the proper connection number of a connected graph equals 1 if and only if is a complete graph by the authors in \cite{Andrews2016, Borozan2012}, our motivation is to characterize 2-connected graphs which have proper connection number 2. First of all, we disprove Conjecture 3 in \cite{Borozan2012} by constructing classes of 2-connected graphs with minimum degree that have proper connection number 3. Furthermore, we study sufficient conditions in terms of the ratio between the minimum degree and the order of a 2-connected graph implying that has proper connection number 2. These results are presented in Chapter 4 of the dissertation.
In Chapter 5, we study proper connection number at most 2 of connected graphs in the terms of connectivity and forbidden induced subgraphs , where are three integers and (where is the graph consisting of three paths with and edges having an end-vertex in common).
Recently, there are not so many results on the proper -connection number , where is an integer. Hence, in Chapter 6, we consider the proper 2-connection number of several classes of connected graphs. We prove a new upper bound for and determine several classes of connected graphs satisfying . Among these are all graphs satisfying the Chv\'{a}tal and Erd\'{o}s condition ( with two exceptions). We also study the relationship between proper 2-connection number and proper connection number of the Cartesian product of two nontrivial connected graphs.
In the last chapter of the dissertation, we propose some open problems of the proper connection number and the proper 2-connection number
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
Claw-free t-perfect graphs can be recognised in polynomial time
A graph is called t-perfect if its stable set polytope is defined by
non-negativity, edge and odd-cycle inequalities. We show that it can be decided
in polynomial time whether a given claw-free graph is t-perfect
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
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