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On the longest path in a recursively partitionable graph

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A connected graph G with order n ≥ 1 is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to K1, or for every sequence (n1, . . . , np) of positive integers summing up to n there exists a partition (V1, . . . , Vp) of V (G) such that each Vi induces a connected R-AP subgraph of G on ni vertices. Since previous investigations, it is believed that a R-AP graph should be “almost traceable” somehow. We first show that the longest path of a R-AP graph on n vertices is not constantly lower than n for every n. This is done by exhibiting a graph family C such that, for every positive constant c ≥ 1, there is a R-AP graph in C that has arbitrary order n and whose longest path has order n−c. We then investigate the largest positive constant c’ < 1 such that every R-AP graph on n vertices has its longest path passing through n • c’ vertices. In particular, we show that c’ ≥ 2/3 . This result holds for R-AP graphs with arbitrary connectivity.
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Bibliogr. 8 poz., rys., tab.
  • Univ. Bordeaux LaBRI, UMR 5800, F-33400 Talence, France
  • CNRS LaBRI, UMR 5800, F-33400 Talence, France
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  • [4] O. Baudon, J. Bensmail, F. Foucaud, M. Pilsniak, Structural properties of recursively partitionable graphs with connectivity 2. Preprint available at:
  • [5] O. Baudon, F. Foucaud, J. Przybyło, M. Wozniak, Structure of k-connected arbitrarily partitionable graphs. Preprint available at:
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  • [7] O. Baudon, F. Gilbert, M. Wozniak, Recursively arbitrarily vertex-decomposable graphs, Opuscula Math. 32 (2012) 4, 689–706.
  • [8] A. Marczyk, An ore-type condition for arbitrarily vertex decomposable graphs, Discret. Math. 309 (2009) 11, 3588–3594.
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