The disjoint multiple qA

PDF p. 2: Section 2, paragraph before Lemma 2.1

Let $A=(V(A),E(A))$ be a finite simple graph with at least one vertex, and let $q$ be a positive integer. Define $qA$ to be the graph with vertex set $\{1,\dots ,q\}\times V(A)$ and edge set $\{\{(i,u),(i,v)\}:1\le i\le q,\{u,v\}\in E(A)\}$.

Thus $qA$ is the disjoint union of $q$ copies of $A$: the copy indexed by $i$ has vertices $\{i\}\times V(A)$, and distinct copies have disjoint vertex sets and no edges between them. A copy here means a subgraph isomorphic to $A$; an isomorphism is a bijection of vertex sets that preserves and reflects adjacency.

Graph and colouring conventions

Tagged graph families with anticomplete parts

Apply the tagged graph-family construction to the finite index set $\{1,\dots ,q\}$, taking each indexed graph to be $A$. It gives the displayed finite simple graph, with pairwise disjoint tagged parts and no edges between distinct parts. Within the part indexed by $i$, the map $v\mapsto (i,v)$ is a bijection that preserves and reflects adjacency, so that part with its internal edges is a copy of $A$. Since $q\ge 1$ and $V(A)$ is nonempty, $qA$ has at least one vertex and meets the stated graph convention. The displayed sets determine the graph uniquely; any other disjoint union with the same indexed copies is isomorphic to it by the componentwise copy isomorphisms.