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

Abstract

For a graph G=(V,E) and a set S⊆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)∩E(P2)=0̸ and V(P1)∩V(P2)=S. Let πG(S) denote the maximum number of internally disjoint S-paths in G. The k-path-connectivityπk(G) of G is then defined as the minimum πG(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 Γn. We find that Γn always has a nice structure connecting any 3-subset S of V(Γn), according to the parity of n. Thereby, we show that π3Γn=⌊3n4⌋−1, for any n≥3.