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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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