Elsevier

Discrete Applied Mathematics

Volume 157, Issue 2, 28 January 2009, Pages 330-338
Discrete Applied Mathematics

Fall colouring of bipartite graphs and cartesian products of graphs

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Abstract

The question of whether a graph can be partitioned into k independent dominating sets, which is the same as having a fall k-colouring, is considered. For k=3, it is shown that a graph G can be partitioned into three independent dominating sets if and only if the cartesian product GK2 can be partitioned into three independent dominating sets. The graph K2 can be replaced by any graph H such that there is a mapping f:QnH, where f is a type-II graph homomorphism.

The cartesian product of two trees is considered, as well as the complexity of partitioning a bipartite graph into three independent dominating sets, which is shown to be NP-complete. For other values of k, iterated cartesian products are considered, leading to a result that shows for what values of k the hypercubes can be partitioned into k independent dominating sets.

Keywords

Hypercube
Indominable
Fall colouring
Domatic partition
Idomatic partition
Idomatic number
Cartesian product

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