Labeled Packing of Cycles and Circuits

In 2013, Duch{\^e}ne, 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 $\lambda$ 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 $\ge$ 2 and 1 $\le$ x $\le$ 2k -- 1, the value of $\lambda$ 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.