The κk-connectivity of line graphs

Let S be a set of at least two vertices in a graph G. A subtree T of G is an S-Steiner tree if S subset of V(T). Two S-Steiner trees T-1 and T-2 are internally disjoint (resp. edge-disjoint) if E(T-1)boolean AND E(T-2) = empty set and V(T-1)boolean AND V(T-2) = S (resp. E(T-1)boolean AND E(T-2) = empty set). Let kappa(G)(S) (resp. lambda(G)(S)) be the maximum number of internally disjoint (resp. edge-disjoint) S-Steiner trees in G, and let kappa(k)(G) (resp. lambda(k)(G)) be the minimum kappa(G)(S) (resp. lambda(G)(S)) for S ranges over all k-subsets of V(G). It is well-known that the connectivity of the line graph of a graph G is closely related to the edge-connectivity of G. Chartrand et al., Li et al., and Li et al. showed that kappa(k)(L(G)) >= lambda(k)(G) for k = 2,3,4, respectively. In [Discrete Math. 341(2018), 1945-1951], Li et al. also proposed the following conjecture: kappa(k)(L(G)) >= lambda(k)(G) for integer k >= 2 and a graph G with at least k vertices. In this paper, we show that kappa(k)(L(G)) >= lambda(k)(G) - inverted right perpendicular inverted right perpendicular mad(G)/2inverted left perpendicular/2 inverted left perpendicular for general k, where mad G) is the maximum average degree of a graph G. We also show that kappa(k)(L(G)) >= lambda(k)(G) - inverted right perpendicular left perpendicular r/2 right perpendicular inverted left perpendicular for any integers k and r such that k <= (r2). Finally, we show that the conjecture holds for k = 5, and the conjecture holds for the wheel graph and the complete bipartite graph K-3,K-n. (C) 2020 Elsevier B.V. All rights reserved.