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The graph of labellings of a given graph. (English) Zbl 0665.05041
Let be $$H=(V,E)$$ a finite undirected graph with n vertices. A labelling of H is as known a bijection $$\lambda$$ : V(H)$$\to \{1,...,n\}$$ and there are n! ordered pairs $$(H,\lambda_ i)$$. In the set of these pairs is defined the property of isomorphism, which is an equivalence relation. Any class of this equivalence is called a labelled graph H and the set of all labelled graphs is denoted by $$\Lambda$$ (H). In this paper a graph $$\Pi$$ (H) is considerd whose vertex set consists of $$\Lambda$$ (H), this means consequently that all graphs $$H_ i$$ on the vertex set $$\{$$ 1,...,n$$\}$$ isomorphic to H are the vertices of this derived graph, and two vertices $$H_ 1$$, $$H_ 2$$ are adjacent iff holds $| E(H_ 1)-E(H_ 2)| =| E(H_ 2)-E(H_ 1)| =1.$ It is shown that $$\Pi$$ (H), which arises from a regular graph H, has no edges and this means $$\Pi$$ (H) is discrete (Theorem 1). By four further theorems such classes of graphs H are characterized for which $$\Pi$$ (H) is disconnected and in Theorem 6 is proved that holds $$\Pi$$ (H)$$\cong \Pi (\overline{H})$$, where $$\overline{H}$$ is the complement of H.
Finally the author formulates two problems to the characterization of such graphs H for which $$\Pi$$ (H) is discrete respectively disconnected.
Reviewer: H.-J.Presia
##### MSC:
 05C75 Structural characterization of families of graphs
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##### References:
 [1] TOMASTA P.: Problems 15-18. Czechoslovak Conference on Graph Theory and Combinatorics, Racek Valley, May 1986.
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