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    Proper connection number of graphs

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    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 GG 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 PP in an edge-coloured graph GG 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 GG 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 pc(G)pc(G), of a connected graph GG is the smallest number of colours that are needed in order to make GG properly connected. Let k2k\geq2 be an integer. If every two vertices of an edge-coloured graph GG are connected by at least kk proper paths, then GG is said to be a \emph{properly kk-connected graph}. The \emph{proper kk-connection number} pck(G)pc_k(G), introduced by Borozan et al. \cite{Borozan2012}, is the smallest number of colours that are needed in order to make GG a properly kk-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 GG equals 1 if and only if GG 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 δ(G)3\delta(G)\geq3 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 GG implying that GG 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 Si,j,kS_{i,j,k}, where i,j,ki,j,k are three integers and 0ijk0\leq i\leq j\leq k (where Si,j,kS_{i,j,k} is the graph consisting of three paths with i,ji,j and kk edges having an end-vertex in common). Recently, there are not so many results on the proper kk-connection number pck(G)pc_k(G), where k2k\geq2 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 pc2(G)pc_2(G) and determine several classes of connected graphs satisfying pc2(G)=2pc_2(G)=2. Among these are all graphs satisfying the Chv\'{a}tal and Erd\'{o}s condition (α(G)κ(G)\alpha({G})\leq\kappa(G) with two exceptions). We also study the relationship between proper 2-connection number pc2(G)pc_2(G) and proper connection number pc(G)pc(G) 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

    Proper Hamiltonian Cycles in Edge-Colored Multigraphs

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    A cc-edge-colored multigraph has each edge colored with one of the cc available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree.Comment: 13 page
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