## 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

dmtcs:1259 - Discrete Mathematics & Theoretical Computer Science, April 15, 2014, Vol. 16 no. 1 - https://doi.org/10.46298/dmtcs.1259
On the Cartesian product of of an arbitrarily partitionable graph and a traceable graph

Authors: Olivier Baudon ; Julien Bensmail ; Rafał Kalinowski ; Antoni Marczyk ; Jakub Przybyło ; Mariusz Wozniak

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∈ &#x007b;1,\textellipsis,k&#x007d;, 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.

Volume: Vol. 16 no. 1
Section: Graph Theory
Published on: April 15, 2014
Accepted on: July 23, 2015
Submitted on: November 23, 2012
Keywords: Discrete Mathematics,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]