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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
On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
Courant theorem provides an upper bound for the number of nodal domains of
eigenfunctions of a wide class of Laplacian-type operators. In particular, it
holds for generic eigenfunctions of quantum graph. The theorem stipulates that,
after ordering the eigenvalues as a non decreasing sequence, the number of
nodal domains of the -th eigenfunction satisfies . Here,
we provide a new interpretation for the Courant nodal deficiency in the case of quantum graphs. It equals the Morse index --- at a
critical point --- of an energy functional on a suitably defined space of graph
partitions. Thus, the nodal deficiency assumes a previously unknown and
profound meaning --- it is the number of unstable directions in the vicinity of
the critical point corresponding to the -th eigenfunction. To demonstrate
this connection, the space of graph partitions and the energy functional are
defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure
On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor
The generalised colouring numbers and
were introduced by Kierstead and Yang as a generalisation
of the usual colouring number, and have since then found important theoretical
and algorithmic applications. In this paper, we dramatically improve upon the
known upper bounds for generalised colouring numbers for graphs excluding a
fixed minor, from the exponential bounds of Grohe et al. to a linear bound for
the -colouring number and a polynomial bound for the weak
-colouring number . In particular, we show that if
excludes as a minor, for some fixed , then
and
.
In the case of graphs of bounded genus , we improve the bounds to
(and even if
, i.e. if is planar) and
.Comment: 21 pages, to appear in European Journal of Combinatoric
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