Rankings of graphs

Abstract

A vertex (edge) coloring c∶V → {1, 2, ⋯, t} (c′∶E → {1, 2, ⋯, t}) of a graph G=(V, E) is a vertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number χ _r (G) (edge ranking number $\chi '_r \left( G \right)$) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the VERTEX RANKING and EDGE RANKING problems. Among others it is shown that χ _r (G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k. We characterize those graphs where the vertex ranking number χ _r and the chromatic number χ coincide on all induced subgraphs, show that χ _r (G)=χ(G) implies χ(G)=ω(G) (largest clique size) and give a formula for $\chi '_r \left( {K_n } \right)$.

This author was partially supported by the ESPRIT Basic Research Actions of the EC under contract 7141 (project ALCOM II)

This author was partially supported by the Office of Naval Research under Grant No. N0014-91-J-1693

Research of this author was done while he visited IRISA, Rennes Cedex, France. This author was partially supported by Deutsche Forschungsgemeinschaft under Kr 1371/1-1

This author was partially supported by the “OTKA” Research Fund of the Hungarian Academy of Sciences, Grant No. 2569