First-Fit Coloring of Forests in Random Arrival Model

TL;DR

This work analyzes First-Fit online coloring on forests with random vertex arrival. It proves a tight threshold for the expected color usage, showing $(1/2 \pm o(1)) \\cdot \\frac{\\ln n}{\\ln\\ln n}$ colors, with matching upper and lower bounds derived via a bidirected-path framework and a hierarchical tree construction under a two-stage random model. The techniques connect color counts to bidirected path lengths in the oriented graph and exploit probabilistic bounds to control the color distribution. The results reveal a meaningful average-case improvement over adversarial bounds and motivate extensions to bipartite and more general graph classes, with potential implications for distributed coloring schemes.

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.

Theorems & Definitions (6)

• Theorem 1
• Corollary 3
• Lemma 4
• Corollary 5
• Lemma 6
• Conjecture 11