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# Chaînes alternées dans les graphes arête-coloriés : k-linkage et arbres couvrants

Abstract : A c-edge-colored graph Gc is a graph whose edges are colored by a given set of colors. A subgraph of Gc is proper if no two adjacent edges have the same color.A c-edge-colored graph or multigraph Gc is k-linked (respectively k-edge-linked) if for any 2k distinct vertices, say x1 y1 , x2 y2 , ..., xk yk , there exist k vertex-disjoint (respectively edge-disjoint) proper paths joining x1 to y1 , x2 to y2 , ... , xk to yk .A proper spanning tree of a graph Gc is a spanning tree such that any two adjacent edges differ in colors.A weak spanning tree is a spanning rooted tree such that there exists a proper path between the root and every vertex of the graph.In the first part of this thesis, we provide conditions which are sufficient for an edge-colored graph to be k-linked. It is a classic problem in graph theory , with many applications. So, we established among others the following results.A) Every 2-edge-colored multigraph of order n ≥ 242k such that dc(Gc) ≥ n/2+k –1, is k-linked.B) Every c-edge-colored multigraph of order n ≥ 2k and size m≥ cn(n–1)/2 – c(n–2k +1)+1 is k-linked.C) Every c-edge-colored multigraph of order n ≥ 2k is k-edge-linked if for each vertex x, dc(x) ≥ n/2.D) Every 2-edge-colored multigraph of order n ≥ 2k ≥ 10 and size m ≥ n2 – 5n + 11 is k-edge-linked if for each vertex x, dc(x) ≥ 1.In the second part of this thesis, two other classic problems in graph theory are treated in edge-colored version: spanning trees and hamiltonian paths. We give below some results.E) Every c-edge-colored simple k-connected graph of order n ≥ C²k+1 + k + 2 with c ≥ C²n–k–1 + k +1, has a proper spanning tree.F) Every c-edge-colored connected graph Gc of rainbow degree rd(Gc)=k and order n ≥ C²k+1 + k + 2 with c ≥ C²n–k–1 + k +1, has a proper spanning tree. G) Every c-edge-colored simple k-connected graph of order n ≥ ((k + j)2 + 3(k + j) – 2)/2 and c ≥ ((n – k – j)(n – k – j – 1))/2 + 2 , with j(j –1)=k , has a weak spanning tree.H) Every c-edge-colored multigraph Gc of order n ≥ 14 and size m ≥ (n – 3)(n – 4) + 3n – 2 such that rd(Gc) = 2, has a proper hamiltonian path.I) Every c-edge-colored multigraph of order n ≠ 5, 7 and size m ≥ n2 – 3n + 4, has a proper hamiltonian path.Most of the given results are the best possible with regard to the properties on the sufficient conditions.
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Submitted on : Friday, January 4, 2013 - 1:52:17 AM
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• HAL Id : tel-00769929, version 1

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Gervais Mendy. Chaînes alternées dans les graphes arête-coloriés : k-linkage et arbres couvrants. Autre [cs.OH]. Université Paris Sud - Paris XI; Université Cheikh Anta Diop (Dakar), 2011. Français. ⟨NNT : 2011PA112194⟩. ⟨tel-00769929⟩

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