2024/04/25 by John Iacono, Iacono, John, Piotr Micek +5
Computer Science · #Advanced Algebra and Logic #Advanced Graph Theory Research #Combinatorics (math.CO) #Data Management and Algorithms #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2404.16340
openalex publication_date 2024/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, χℓ-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 χℓ-vr(G)= O(n1-2/(ℓ+1)log n) if ℓ is even and χℓ-vr(G)= O(n1-2/ℓlog n) if ℓ is odd. The case ℓ=2 resolves (up to the log n factor) an open problem posed by \citetkarpas.neiman.ea:on and the cases ℓ∈\2,3\ are asymptotically optimal (up to the log n factor).