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On Borodin-Kostochka conjecture for correspondence coloring

Zdeněk Dvořák, Ross J. Kang, David Mikšaník

Abstract

Borodin and Kostochka in 1977 conjectured that if a graph $G$ has maximum degree $Δ(G)\ge 9$ and its clique number satisfies $ω(G)\le Δ(G)-1$, then its chromatic number satisfies $χ(G) \le Δ(G)-1$. We prove this statement with respect to a stronger graph coloring parameter, the correspondence chromatic number $χ_{DP}$, provided the maximum degree is sufficiently large. More precisely, we prove that for every integer $Δ\ge 3\cdot 10^9$, a graph $G$ of maximum degree at most $Δ$ satisfies $χ_{DP}(G) \le \max(ω(G),Δ-1)$. This strengthens earlier results of Reed (1999) for usual chromatic number and of Choi, Kierstead and Rabern (2023) for list chromatic number.

On Borodin-Kostochka conjecture for correspondence coloring

Abstract

Borodin and Kostochka in 1977 conjectured that if a graph has maximum degree and its clique number satisfies , then its chromatic number satisfies . We prove this statement with respect to a stronger graph coloring parameter, the correspondence chromatic number , provided the maximum degree is sufficiently large. More precisely, we prove that for every integer , a graph of maximum degree at most satisfies . This strengthens earlier results of Reed (1999) for usual chromatic number and of Choi, Kierstead and Rabern (2023) for list chromatic number.
Paper Structure (13 sections, 33 theorems, 60 equations, 1 figure)

This paper contains 13 sections, 33 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Correspondence coloring
  4. Probabilistic tools
  5. Forbidden induced subgraphs
  6. Graphs with many universal vertices
  7. Vertex neighborhoods
  8. Near-clique neighborhoods
  9. Sparse-dense decomposition
  10. Probabilistic analysis
  11. Transversal of the setting
  12. Uniform coloring of the transversal
  13. Putting things together

Key Result

Theorem 2

For every graph $G$ such that $\Delta(G) \geq 10^{14}$, if $\omega(G) \leq \Delta(G) - 1$, then $\chi(G) \leq \Delta(G) - 1$.

Figures (1)

  • Figure 1: The graph $C_5 \boxtimes K_3$.

Theorems & Definitions (74)

  • Conjecture 1: Borodin and Kostochka BK77
  • Theorem 2: Reed Ree99
  • Theorem 3: Choi, Kierstead, and Rabern CKR23
  • Theorem 4
  • proof
  • Theorem 14: Lovász Local Lemma
  • Theorem 15: Chernoff Bounds
  • Theorem 16: McDiarmid's Inequality
  • Lemma 17
  • proof
  • ...and 64 more