The generalized 4-connectivity of complete-transposition graphs

AbstractThe fault tolerability of the network is usually measured by the classical or generalized connectivity of the graph. For any subset S⊆V(G) with |S|≥2, a tree T is called an S-tree if S⊆V(T). Furthermore, any two S-tree T1 and T2 are internally disjoint if E(T1)∩E(T2)=∅ and V(T1)∩V(T2)=S. We denote by κG(S) the maximum number of pairwise internally disjoint S-trees in G. For an integer k≥2, the generalized k-connectivity of a graph G is defined as κk(G)=min{κG(S)|S⊆V(G) and |S|=k}. In this paper, we establish the generalized 4-connectivity of the Cayley graph CTn generated by complete graphs.Keywords: Fault tolerabilitygeneralized connectivityCayley graphscomplete graphs Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partly supported by the National Natural Science Foundation of China (Nos. 61977016, 61572010, and 62277010), Natural Science Foundation of Fujian Province, China (Nos. 2020J01164, 2017J01738). This work was also partly supported by China Scholarship Council (CSC No. 202108350054).