Let G be a simple graph of order $$n\ge 2$$ and let $$k\in \{1,\ldots ,n-1\}$$ . The k-token $$F_k(G)$$ is the whose vertices are k-subsets V(G), where two adjacent in whenever their symmetric difference an edge G. In 2018 Leaños Trujillo-Negrete proved that if t-connected $$t\ge k$$ , then at least $$k(t-k+1)$$ -connected. this paper we show such lower bound remains true context edge-connectiv...

Abstract In this note we examine the following random graph model: for an arbitrary , with quadratic many edges, construct a by randomly adding edges to and coloring of colors. We show that large enough constant every pair vertices in are joined rainbow path, is, is connected high probability. This confirms conjecture Anastos Frieze, who proved statement resolved case when function .

The l-component (edge) connectivity of a graph G, denoted by cκl(G) (cλl(G)), is the minimum number vertices (edges) whose removal from G results in disconnected with at least l components. pancake Pn popular underlying topology for distributed systems. In paper, we determine cκl(Pn) and cλl(Pn) 3 ≤ 5.