Vertex Ranking of Degenerate Graphs
John Iacono, Piotr Micek, Pat Morin, and Bruce Reed

TL;DR
This paper establishes bounds on the vertex-ranking number for degenerate graphs, resolving an open problem for the case 62 and providing asymptotically optimal results for small 6 and 63.
Contribution
It provides new bounds on the 6-vertex-ranking number for degenerate graphs, solving an open problem for 62 and improving understanding of graph coloring complexity.
Findings
For fixed d and 6, the 6-vertex-ranking number is bounded by O(n^{1-2/(6+1)} 7 log n) if 6 is even.
For odd 6, the bound is O(n^{1-2/6} 7 log n).
The case 62 resolves an open problem up to a 7 log n factor.
Abstract
An -vertex-ranking of a graph is a colouring of the vertices of with integer colours so that in any connected subgraph of with diameter at most , there is a vertex in whose colour is larger than that of every other vertex in . The -vertex-ranking number, , of is the minimum integer such that has an -vertex-ranking using colours. We prove that, for any fixed and , every -degenerate -vertex graph satisfies if is even and if is odd. The case resolves (up to the factor) an open problem posed by \citet{karpas.neiman.ea:on} and the cases are asymptotically optimal (up to the factor).
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Taxonomy
TopicsAdvanced Graph Theory Research · Advanced Algebra and Logic · Data Management and Algorithms
