For a graph [Formula: see text] and a set [Formula: see text] of size at least [Formula: see text], a path in [Formula: see text] is said to be an [Formula: see text]-path if it connects all vertices of [Formula: see text]. Two [Formula: see text]-paths [Formula: see text] and [Formula: see text] are said to be internally disjoint if [Formula: see text] and [Formula: see text]. Let [Formula: see text] denote the maximum number of internally disjoint [Formula: see text]-paths in [Formula: see text]. The [Formula: see text]-path-connectivity [Formula: see text] of [Formula: see text] is then defined as the minimum [Formula: see text], where [Formula: see text] ranges over all [Formula: see text]-subsets of [Formula: see text]. In [M. Hager, Path-connectivity in graphs, Discrete Math. 59 (1986) 53–59], the [Formula: see text]-path-connectivity of the complete bipartite graph [Formula: see text] was calculated, where [Formula: see text]. But, from his proof, only the case that [Formula: see text] was considered. In this paper, we calculate the situation that [Formula: see text] and complete the result.
For a graph G=(V,E) and a set S⊆V(G) of a 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)=∅ and V(P1)∩V(P2)=S; that is, they share no vertices and edges apart from S. Let πG(S) denote the maximum number of internally disjoint S-paths in G. The k-path-connectivity πk(G) of G is then defined as the minimum πG(S), where S ranges over all k-subsets of V(G). In this paper, we study the k-path-connectivity of the complete balanced tripartite graph Kn,n,n and obtain πkKn,n,n=2nk−1 for 3≤k≤n.
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