The difference between the chromatic and the cochromatic number of a random graph

2024/09/26 by Annika Heckel, Heckel, Annika
Computer Science · Mathematics · Physics and Astronomy · #05C15 #05C80 #Combinatorics (math.CO) #Complex Network Analysis Techniques #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.2409.17614

openalex publication_date 2024/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The cochromatic number ζ(G) of a graph G is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let χ(G) denote the usual chromatic number. Around 1991 Erdős and Gimbel asked: For the random graph G ∼ G_{n, 1/2}, does χ(G)-ζ(G) → ∞ whp? Erdős offered $100 for a positive and $1,000 for a negative answer. We give a positive answer to this question for roughly 95% of all values n.

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