First-Fit Coloring of Forests in Random Arrival Model

TL;DR

This paper analyzes the First-Fit coloring algorithm on forests under a random vertex arrival model, establishing an asymptotically optimal bound on the number of colors used.

Contribution

It proves an asymptotically tight bound on the expected number of colors for First-Fit coloring of forests in a random order setting.

Findings

Expected colors used is at most (1+o(1))·ln n / ln ln n

Constructed forests show the bound is tight

First-Fit is near-optimal for forests in this model

Abstract

We consider a graph coloring algorithm that processes vertices in order taken uniformly at random and assigns colors to them using First-Fit strategy. We show that this algorithm uses, in expectation, at most $(1 + o(1))\cdot \ln n \,/\, \ln\ln n$ different colors to color any forest with $n$ vertices. We also construct a family of forests that shows that this bound is best possible.