Abstract
Graphs and Algorithms The grundy numbering of a graph is the maximum number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices. The grundy numbering problem is to find this ordering. We prove that there is a constant c>1 so that approximating the grundy numbering problem within c is not possible, unless NP ⊆ RP
Highlights
In first-fit coloring, we color the vertices with positive integers
The grundy number of a graph is the maximum possible number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices
In Kierstead (1988) the author studies the grundy number problem on interval graphs and establishes that the grundy number differs from the obvious lower bound of maximum clique size by at most some multiplicative universal constant
Summary
In first-fit coloring, we color the vertices with positive integers. The vertices arrive on-line, and we color the vertex v by the minimum number not appearing in its neighborhood. The grundy number of a graph is the maximum possible number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices. In Kierstead (1988) the author studies the grundy number problem on interval graphs and establishes that the grundy number differs from the obvious lower bound of maximum clique size by at most some multiplicative universal constant. An algorithm solving this problem on trees is given in Hedetniemi et al (1982) and on partial k-trees in Telle and Proskurowski (1997). There was no hardness of approximation result for the problem
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
