Some properties of transposition graphs

Abstract

For every finite graph G without isolated vertices, there is an associated set of transpositions Ω(G) which correspond in a natural way to the edges of G. G. Ω(G) generates a group H which is a symmetric group iff G is connected. The Cayley graph Γ(H,Ω) clearly depends only on G, and is called the transposition graph of G,.r(G).

The distance between any two vertices of a transposition graph Γ(G) is established in the cases where G is a complete graph, a complete graph with an edge deleted, a path graph, or a star. The diameter of Γ(G) is obtained as a corollary in these cases. General upper and lower bounds are found for the diameter of Γ(G) which depend on the number of vertices and the diameter of G.

If G has no connected components isomorphic to C₄ or K_n then the automorphisms of Γ(G) are completely determined by the automorphisms of G. In particular, if G is a connected graph on n vertices with no non-trivial automorphisms, then Γ(G) is a graphical regular representation of S_n. 

Every transposition graph with at least four vertices is hamiltonian. 

If the complement of the line graph of a graph G is hamiltonian then the genus of Γ(G) depends only on the number of vertices and edges of G. This result can be generalised if G has no circuits of length three. 

Finally, it is proved that the Complement of the line graph of a graph G is hamiltonian if every vertex of G is incident to at most half the edges of 0 and every edge of G is non-incident to at least two other edges of G, provided G has at least thirty four edges.

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