[Paper Review] A survey of $\chi$-boundedness

Alex Scott, Paul Seymour|arXiv (Cornell University)|Dec 18, 2018

Limits and Structures in Graph Theory99 references18 citations

TL;DR

This survey comprehensively reviews recent advances in $χ$-boundedness, a central concept in graph theory concerning the relationship between chromatic number ($\chi$) and clique number ($\omega$). It synthesizes results on $χ$-bounded ideals—particularly those excluding odd holes, cycles with chords, or specific vertex-minor structures—providing tight bounds and highlighting open problems, including polynomial $χ$-binding functions and algorithmic implications.

ABSTRACT

If a graph has bounded clique number and sufficiently large chromatic number, what can we say about its induced subgraphs? András Gyárfás made a number of challenging conjectures about this in the early 1980s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.

Motivation & Objective

• To systematize and summarize recent breakthroughs in $χ$-boundedness, a long-standing area of research initiated by András Gyárfás.
• To clarify the structural conditions under which graphs with bounded clique number must have chromatic number bounded by a function of the clique number.
• To identify and analyze key conjectures and theorems related to $χ$-bounded ideals, especially those excluding odd holes, cycles with chords, or specific vertex-minor configurations.
• To highlight open problems and algorithmic challenges in computing chromatic number and detecting induced substructures in $χ$-bounded classes.

Proposed method

• Surveying foundational results, including the strong perfect graph theorem and the Erdős–Lovász–Tutte constructions of triangle-free graphs with high chromatic number.
• Analyzing structural decompositions and forbidden subgraph characterizations to prove $χ$-boundedness in various graph classes.
• Applying techniques from extremal graph theory, random graph methods, and explicit constructions (e.g., Mycielski, Erdős, Tutte) to demonstrate the existence of graphs with high $χ$ and low $ω$.
• Using vertex-minor operations and circle graph representations to establish $χ$-boundedness in closed ideals under vertex-minor closure.
• Leveraging algorithmic results such as polynomial-time odd hole detection to support constructive bounds on $χ$ in terms of $ω$.
• Presenting conjectures on polynomial $χ$-binding functions and structural partitioning (e.g., partitioning into perfect graphs) to strengthen known bounds.

Experimental results

Research questions

• RQ1 Under what conditions does a graph with bounded clique number and large chromatic number necessarily contain an induced odd hole?
• RQ2 Can the chromatic number of $H$-free graphs be bounded by a function of the clique number for all $H$?
• RQ3 Is the ideal of graphs excluding a fixed number of chords on a cycle $χ$-bounded?
• RQ4 Are all proper ideals closed under vertex-minors $χ$-bounded, and can maximum clique and stable set be computed in polynomial time in such classes?
• RQ5 Can $χ$-binding functions be made polynomial rather than doubly exponential in $ω$ for key graph classes?

Key findings

• For every $\kappa$, if $\omega(G) \leq \kappa$ and $\chi(G) > 2^{2^{\kappa+2}}$, then $G$ contains an odd hole, confirming a conjecture of Gyárfás.
• The ideal of graphs with no odd hole is $χ$-bounded with a doubly exponential $χ$-binding function $f(x) = 2^{2^{x+2}}$.
• Graphs excluding cycles with exactly $k$ chords as induced subgraphs are $χ$-bounded for $k = 2, 3$, and a conjecture extends this to all $k$.
• The ideal of graphs with no induced cycle having a vertex with $k$ neighbors on the cycle is conjectured to be $χ$-bounded.
• The ideal of circle graphs is polynomially $χ$-bounded with a quadratic binding function, and this extends to ideals closed under vertex-minors excluding all circle graphs.
• Recent results confirm that every proper ideal closed under vertex-minors is $χ$-bounded, supporting a long-standing conjecture by Jim Geelen.