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.

Source : oai:HAL:hal-01179212v1

Volume: Vol. 16 no. 1 (in progress)

Section: Graph Theory

Published on: April 15, 2014

Submitted on: November 23, 2012

Keywords: Discrete Mathematics,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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