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Graphs and colors : edge-colored graphs, edge-colorings and proper connections

Abstract : 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(n⁴) 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∆-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(n⁴) known algorithm.
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Leandro Pedro Montero. Graphs and colors : edge-colored graphs, edge-colorings and proper connections. Other [cs.OH]. Université Paris Sud - Paris XI, 2012. English. ⟨NNT : 2012PA112368⟩. ⟨tel-00776899⟩



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