Dans cette thèse nous étudions différents problèmes de graphes et multigraphes arêtes-coloriés tels que la connexité propre, la coloration forte d'arêtes et les chaînes et cycles hamiltoniens propres. Enfin, nous améliorons l'algorithme connu $O(n^4)$ pour décider du comportement d'un graphe sous opérateur biclique, en étudiant les bicliques dans les graphes sans faux jumeaux. Plus précisément, 1) Nous étudions d'abord le nombre $k$-connexité-propre des graphes, noté $pc_k(G)$, ç'est à dire le nombre minimum de couleurs nécessaires pour colorer les arêtes d'un graphe de façon à ce qu'entre chaque paire de sommets, ils existent $k$ chemins intérieurement sommet-disjoints. Nous prouvons plusieurs bornes supérieures pour $pc_k(G)$. Nous énonçons quelques conjectures pour les graphes généraux et bipartis et nous les prouvons dans le cas où $k = 1$. 2) Nous étudions l'existence de chaînes et de cycles hamiltoniens propres dans les multigraphes arêtes-coloriés. Nous établissons des conditions suffisantes, en fonction de plusieurs paramètres tels que le nombre d'arêtes, le degré arc-en-ciel, la connexité, etc. 3) Nous montrons que l'indice chromatique fort est linéaire au degré maximum pour tout graphe $k$-dégénéré où, $k$ est fixe. En corollaire, notre résultat conduit à une amélioration des constantes et donne également un algorithme plus simple et plus efficace pour cette famille de graphes. De plus, nous considérons les graphes planaires extérieurs. Nous donnons une formule pour trouver l'indice chromatique fort exact pour les graphes bipartis planaires extérieurs. Nous améliorons également la borne supérieure pour les graphes planaires extérieurs généraux. 4) Enfin, nous étudions les bicliques dans les graphes sans faux jumeaux et nous présentons ensuite un algorithme $O(n+m)$ pour reconnaître les graphes convergents et divergents en améliorant l'algorithme $O(n^4)$. / In this thesis, we study different problems in edge-colored graphs and edge-colored multigraphs, such as proper connection, strong edge colorings, and proper hamiltonian paths and cycles. Finally, we improve the known $O(n^4)$ algorithm to decide the behavior of a graph under the biclique operator, by studying bicliques in graphs withoutfalse-twin vertices. In particular: 1) We first study the $k$-proper-connection number of graphs, this is, the minimum number of colors needed to color the edges of a graph such that between any pair of vertices there exist $k$ internally vertex-disjoint paths. We denote this number $pc_k(G)$. We prove several upper bounds for $pc_k(G)$. We state some conjectures for general and bipartite graphs, and we prove all of them for the case $k=1$. 2) Then, we study the existence of proper hamiltonian paths and proper hamiltonian cycles in edge-colored multigraphs. We establish sufficient conditions, depending on several parameters such as the number of edges, the rainbow degree, the connectivity, etc. 3) Later, we showthat the strong chromatic index is linear in the maximum degree for any $k$-degenerate graph where $k$ is fixed. As a corollary, our result leads to considerable improvement of the constants and also gives an easier and more efficient algorithm for this familly of graphs. Next, we consider outerplanar graphs. We give a formula to find exact strong chromatic index for bipartite outerplanar graphs. We also improve the upper bound for general outerplanar graphs from the $3\Delta-3$ bound. 4) Finally, we study bicliques in graphs without false-twin vertices and then we present an $O(n+m)$ algorithm to recognize convergent and divergent graphs improving the $O(n^4)$ known algorithm.

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