The generalized 4-connectivity of complete-transposition graphs.

The fault tolerability of the network is usually measured by the classical or generalized connectivity of the graph. For any subset $ S \subseteq V\left (G \right) $ S ⊆ V (G) with $ \left | S \right | \ge 2 $ | S | ≥ 2 , a tree T is called an S-tree if $ S \subseteq V\left (T \right) $ S ⊆ V (T) . Furthermore, any two S-tree $ T_1 $ T 1 and $ T_2 $ T 2 are internally disjoint if $ E\left ({{T_1}} \right) \cap E\left ({{T_2}} \right) = \emptyset $ E ( T 1 ) ∩ E ( T 2 ) = ∅ and $ V\left ({{T_1}} \right) \cap V\left ({{T_2}} \right) = S $ V ( T 1 ) ∩ V ( T 2 ) = S. We denote by $ {\kappa _G}\left (S \right) $ κ G (S) the maximum number of pairwise internally disjoint S-trees in G. For an integer $ k \ge 2 $ k ≥ 2 , the generalized k-connectivity of a graph G is defined as $ {\kappa _k}\left (G \right) = \min \left \{ {{\kappa _G}\left (S \right) | S \subseteq V\left (G \right)} \right. $ κ k (G) = min { κ G (S) | S ⊆ V (G) and $ \left. {\left | S \right | = k} \right \} $ | S | = k } . In this paper, we establish the generalized 4-connectivity of the Cayley graph $ CT_n $ C T n generated by complete graphs. [ABSTRACT FROM AUTHOR]