On the \rho-subdivision number of graphs

Abstract

For an arbitrary invariant ρ(G) of a graph G the ρ-subdivision number sd ρ (G) is the minimum number of edges of G whose subdivision results in a graph H with ρ(H) = ρ(G). Set sd ρ (G) = |E(G)| if such an edge set does not exist.In the first part of this paper we give some general results for the ρsubdivision number. In the second part we study this parameter for the chromatic number, for the chromatic index, and for the total chromatic number. We show among others that there is a strong relationship to the ρedge stability number for these three invariants. In the last part we consider a modification, namely the ρ-multiple subdivision number where we allow multiple subdivisions of the same edge.