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On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

Abstract : A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence τ=(n1,\textellipsis,nk) of positive integers that sum up to n, there exists a partition (V1,\textellipsis,Vk) of the vertex set V(G) such that each set Vi induces a connected subgraph of order ni. A graph G is called AP+1 if, given a vertex u∈V(G) and an index q∈ {1,\textellipsis,k}, such a partition exists with u∈Vq. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G□ H is AP+1. We also prove that G□H is AP, whenever G and H are AP and the order of one of them is not greater than four.
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https://hal.inria.fr/hal-01179212
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Olivier Baudon, Julien Bensmail, Rafał Kalinowski, Antoni Marczyk, Jakub Przybyło, et al.. On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no. 1 (in progress) (1), pp.225--232. ⟨hal-01179212⟩

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