2 research outputs found
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