3-path-connectivity of Cayley graphs generated by transposition trees

For a graph G = (V, E) and a set S & SUBE; V(G) of size at least 2, a path in G is said to be an S -path if it connects all vertices of S. Two S-paths P1 and P2 are said to be internally disjoint if E(P1) & AND; E(P2) = null and V(P1) & AND; V(P2) = S. Let & pi;G(S) denote the maximum number of internally disjoint S-paths in G. The k -path-connectivity & pi;k(G) of G is then defined as the minimum & pi;G(S), where S ranges over all k-subsets of V(G). Cayley graphs often make good models for interconnection networks. In this paper, we consider the 3-path-connectivity of Cayley graphs generated by transposition trees & UGamma;n. We find that & UGamma;n always has a nice structure connecting any 3-subset S of V(& UGamma;n), according to the parity of n. Thereby, we show that & pi;3 (& UGamma;n) = L3n4 <SIC> RIGHT FLOOR - 1, for any n & GE; 3. & COPY; 2023 Elsevier B.V. All rights reserved.