[Paper Review] Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyarfas' conjectures

Maria Chudnovsky, Paul Seymour+1 more|arXiv (Cornell University)|Nov 24, 2014

Advanced Graph Theory Research4 references4 citations

TL;DR

This paper makes progress toward Gyárfás' conjecture on graphs with large chromatic number but no large cliques or long odd holes. It proves that triangle-free graphs with no odd holes longer than ℓ have chromatic number bounded by a function of ℓ, and establishes that pentagonal graphs (with only 5-cycles as odd holes) are 58,000-colorable. The results rely on structural decomposition via level sets in distance partitions and lollipop lemmas to bound chromatic number in induced subgraphs.

ABSTRACT

Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1) Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five; (2) For all $l$, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than $l$; (3) For all $k$, $l$, every graph with no clique of size more than $k$ and sufficiently large chromatic number contains either a 5-hole or a hole of length more than $l$.

Motivation & Objective

• To prove weaker forms of Gyárfás' conjecture on chromatic number boundedness in graphs with restricted clique and odd hole sizes.
• To establish that triangle-free graphs with no odd holes longer than ℓ have chromatic number bounded by a function of ℓ.
• To prove that pentagonal graphs—those with only 5-cycles as induced odd cycles—are 58,000-colorable.
• To develop structural tools, such as lollipop lemmas and level-set analysis, to reduce chromatic number problems to bounded substructures.
• To prove special cases of the conjectures when additional restrictions (e.g., no 5-holes) are imposed.

Proposed method

• Uses a variant of a theorem from previous work to show that if a triangle-free graph has large chromatic number and no long odd holes, then it contains an induced subgraph with a central vertex whose distance levels are mostly stable except one with high chromatic number.
• Applies a lollipop lemma to analyze induced subgraphs with a central component and a path-like tail, using cleanliness to control neighbor structures.
• Employs distance partitioning from a central vertex to decompose the graph into level sets, analyzing chromatic number per level to bound the overall chromatic number.
• Uses induction on the clique number ω(G), proving that if all subgraphs with smaller clique number are colorable, then the full graph is colorable under the given constraints.
• Applies a key lemma to show that if every vertex has neighborhood and second neighborhood with bounded chromatic number, then the whole graph has chromatic number bounded by a function of ℓ and the neighborhood bounds.
• Proves that under the absence of 5-holes and long holes, the chromatic number is bounded by (2ℓ−2)^{2^{ω(G)−1}−1}, using inductive coloring and neighborhood structure analysis.

Experimental results

Research questions

• RQ1 Does every triangle-free graph with no odd hole of length more than ℓ have chromatic number bounded by a function of ℓ?
• RQ2 Can the chromatic number of pentagonal graphs (only 5-cycles as odd holes) be bounded, and if so, what is the bound?
• RQ3 Does the absence of 5-holes and long holes (≤ℓ) imply a chromatic number bound depending on ℓ and the clique number?
• RQ4 Can structural decomposition via distance levels from a central vertex reduce the chromatic number problem to bounded subproblems?
• RQ5 What is the quantitative bound on chromatic number for graphs with no 5-holes and no holes longer than ℓ?

Key findings

• Every pentagonal graph is 58,000-colorable, providing a quantitative bound for the first open case of Gyárfás' conjecture when k=2 and ℓ=5.
• For every ℓ, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length greater than ℓ.
• Every triangle-free graph with no odd hole of length more than ℓ has chromatic number bounded by a function of ℓ, provided that the 2-neighborhood of every vertex has bounded chromatic number.
• If a graph has no 5-hole and no hole of length more than ℓ, then its chromatic number is at most (2ℓ−2)^{2^{ω(G)−1}−1}, a tight bound depending on ℓ and the clique number.
• The proof of the bound for pentagonal graphs relies on showing that any such graph with large chromatic number contains an induced subgraph where every vertex's 2-neighborhood has chromatic number at most 5.
• The paper establishes that conjecture 1.2 (no long holes) holds for triangle-free graphs with no 5-holes, and conjecture 1.3 (no long odd holes) holds for triangle-free graphs with no 5-holes, under the given bounds.