Vertex Ranking of Degenerate Graphs

An ℓ-vertex-ranking of a graph G is a colouring of the vertices of G with integer colours so that in any connected subgraph H of G with diameter at most ℓ, there is a vertex in H whose colour is larger than that of every other vertex in H. The ℓ-vertex-ranking number, χ_ℓ-vr(G), of G is the minimum integer k such that G has an ℓ-vertex-ranking using k colours. We prove that, for any fixed d and ℓ, every d-degenerate n-vertex graph G satisfies χ_ℓ-vr(G)= O(n^1-2/(ℓ+1)log n) if ℓ is even and χ_ℓ-vr(G)= O(n^1-2/ℓlog n) if ℓ is odd. The case ℓ=2 resolves (up to the log n factor) an open problem posed by and the cases ℓ∈{2,3} are asymptotically optimal (up to the log n factor).