Transversal Panconnectedness in Graph Collections

Given a collection G =₁,G₂,,Gₘ\ of graphs on the common vertex set V of size n, a graph H with vertices in V is rainbow in G if there exists an injection φ :E(H)→ [m] such that e ∈ E(Gφ(e)) for all e∈ E(H). We say H is {a} transversal in G if $|E(H)|=m$. Denote δ(G):=min\δ(Gᵢ): i∈ [m]\. For vertices x,\,y ∈ V, let dG(x,y) be the length of the shortest rainbow path (if exists) connecting x and y in G. We say G is transversal panconnected if for any two vertices x, y ∈ V, there exists a rainbow path on k vertices inside G joining x and y for every integer k∈[dG(x,y)+1, n]. In this paper, we provide a tight bound on δ(G) to ensure that G is transversal panconnected. It generalizes the result of [Williamson, Period. Math. Hungar., 1977] to the transversal version and improves the result of [Li, Li and Li, J. Graph Theory, 2023].