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Preprints, Working Papers, ... Year : 2018

Labeled Packing of Cycles and Circuits

Alice Joffard

Abstract

In 2013, Duchêne, Kheddouci, Nowakowski and Tahraoui [4, 9] introduced a labeled version of the graph packing problem. It led to the introduction of a new parameter for graphs, the k-labeled packing number λ k. This parameter corresponds to the maximum number of labels we can assign to the vertices of the graph, such that we will be able to create a packing of k copies of the graph, while conserving the labels of the vertices. The authors intensively studied the labeled packing of cycles, and, among other results, they conjectured that for every cycle C n of order n = 2k + x, with k ≥ 2 and 1 ≤ x ≤ 2k − 1, the value of λ k (C n) was 2 if x was 1 and k was even, and x + 2 otherwise. In this paper, we disprove this conjecture by giving a counter example. We however prove that it gives a valid lower bound, and we give sufficient conditions for the upper bound to hold. We then give some similar results for the labeled packing of circuits.
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Dates and versions

hal-01791213 , version 1 (14-05-2018)

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Alice Joffard, Hamamache Kheddouci. Labeled Packing of Cycles and Circuits. 2018. ⟨hal-01791213⟩
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