Unstable graphs and packing into fifth power

Abstract

In a graph [Formula: see text], a subset [Formula: see text] of the vertex set [Formula: see text] is a module (or interval, clan) of [Formula: see text] if every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. The empty set, the singleton sets, and the full set of vertices are trivial modules. The graph [Formula: see text] is indecomposable (or prime) if all its modules are trivial. If [Formula: see text] is indecomposable, we say that an edge [Formula: see text] of [Formula: see text] is a removable edge if [Formula: see text] is indecomposable (here [Formula: see text]). The graph [Formula: see text] is said to be unstable if it has no removable edges. For a positive integer [Formula: see text], the [Formula: see text]th power [Formula: see text] of a graph [Formula: see text] is the graph obtained from [Formula: see text] by adding an edge between all pairs of vertices of [Formula: see text] with distance at most [Formula: see text]. A graph [Formula: see text] of order [Formula: see text] (i.e., [Formula: see text]) is said to be [Formula: see text]-placeable into [Formula: see text], if [Formula: see text] contains [Formula: see text] edge-disjoint copies of [Formula: see text]. In this paper, we answer a question, suggested by Boudabbous in a personal communication, concerning unstable graphs and packing into their fifth power as follows: First, we give a characterization of the unstable graphs which is deduced from the results given by Ehrenfeucht, Harju and Rozenberg (the theory of [Formula: see text]-structures: a framework for decomposition and transformation of graphs). Second, we prove that every unstable graph [Formula: see text] is [Formula: see text]-placeable into [Formula: see text].