[Paper Review] Bounding $\chi$ by a fraction of $\Delta$ for graphs without large cliques

Marthe Bonamy, Tom Kelly+2 more|arXiv (Cornell University)|Mar 2, 2018

Advanced Graph Theory Research14 references7 citations

TL;DR

This paper establishes a linear improvement over the greedy coloring bound for graphs without large cliques, proving that the list-chromatic number χℓ(G) is O(∆ / √(ln ω / ln ∆)) for graphs with maximum degree ∆ and no clique of size ω. The result holds via a local coloring condition where each vertex’s available colors depend on its degree and neighborhood clique structure, unifying and strengthening prior bounds by Johansson and Reed, and confirming a conjectured linear improvement for large ∆.

ABSTRACT

The greedy coloring algorithm shows that a graph of maximum degree at most $\Delta$ has chromatic number at most $\Delta + 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $\Delta \geq 3$ and the graph contains no clique of size $\Delta + 1$. Reed's Conjecture states that the "greedy bound" can be improved by $k$ if the graph contains no clique of size $\Delta + 1 - 2k$. Johansson proved that the "greedy bound" can be improved by a factor of $\Omega(\ln(\Delta)^{-1})$ or $\Omega\left(\frac{\ln(\ln(\Delta))}{\ln(\Delta)} ight)$ for graphs with no triangles or no cliques of any fixed size, respectively. Notably missing is a $ extit{linear}$ improvement on the "greedy bound" for graphs without large cliques. In this paper, we prove that for sufficiently large $\Delta$, if $G$ is a graph with maximum degree at most $\Delta$ and no clique of size $\omega$, then $$\chi(G) \leq 72\Delta\sqrt{\frac{\ln(\omega)}{\ln(\Delta)}}.$$ This implies that for sufficiently large $\Delta$, if $\omega^{(72c)^2} \leq \Delta$ then $\chi(G) \leq \Delta/c$. This bound actually holds for the list-chromatic and even the correspondence-chromatic number (also known as the DP-chromatic number). In fact, we prove what we call a "local version" of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. Our result simultaneously implies the linear improvement over the "greedy bound" and the two aforementioned results of Johansson.

Motivation & Objective

• To resolve whether the greedy bound on chromatic number can be improved by a linear factor for graphs without large cliques.
• To unify and strengthen prior results by Johansson on triangle-free and H-free graphs.
• To establish a local coloring condition where color availability depends on vertex degree and neighborhood clique size.
• To confirm a conjectured linear improvement over the greedy bound, particularly for graphs with ω = o(∆) and large ∆.
• To extend results to list-chromatic and correspondence chromatic numbers (DP-coloring).

Proposed method

• Introduces a local coloring condition where each vertex v must have |L(v)| ≥ 72 deg(v) · min{ log²(χ(v)+1)/ln(deg(v)), ω(v) ln(ln(deg(v)))/ln(deg(v)), √(ln(ω(v))/ln(deg(v))) } available colors.
• Applies a refined analysis of independent set sizes in neighborhood subgraphs using Ramsey-theoretic bounds and Shearer's lemma.
• Employs Stirling's approximation and logarithmic inequalities to bound the number of independent sets and ensure color availability.
• Uses a probabilistic method with iterative color rejection to show existence of valid colorings under the local condition.
• Leverages results from Molloy and Bernshteyn on independent set lower bounds in graphs without large cliques.
• Proves the main result via Theorem 1.13, which guarantees L-colorability under the derived local conditions.

Experimental results

Research questions

• RQ1 Can the greedy bound of ∆+1 on chromatic number be improved by a linear factor for graphs without large cliques?
• RQ2 Is there a uniform bound on the list-chromatic number that scales as O(∆ / √(ln ω / ln ∆)) for graphs with no clique of size ω?
• RQ3 Can Johansson’s bounds for triangle-free and H-free graphs be unified and generalized under a single local coloring condition?
• RQ4 Does the correspondence chromatic number (DP-chromatic number) also satisfy this improved bound?
• RQ5 What is the best possible function f(c) such that χℓ(G) ≤ ∆/c whenever ω ≤ ∆^{1/f(c)}?

Key findings

• The list-chromatic number χℓ(G) is bounded by O(∆ / √(ln ω / ln ∆)) for graphs with maximum degree ∆ and no clique of size ω.
• For sufficiently large ∆, if ω ≤ (∆^{1/(72c)^2}), then χℓ(G) ≤ ∆/c, confirming a linear improvement over the greedy bound.
• The bound holds not only for chromatic number but also for the list-chromatic and correspondence chromatic numbers (DP-chromatic number).
• A local coloring condition is proven sufficient: each vertex v needs at least 72 deg(v) times the minimum of three terms involving its neighborhood’s chromatic number, clique size, and degree.
• The result unifies and strengthens Johansson’s bounds: it recovers the O(∆ / ln ∆) bound for triangle-free graphs and the O(∆ ln ln ∆ / ln ∆) bound for H-free graphs.
• The paper provides a lower bound on the function f(c) in Question 1.5, showing f(c) ≥ c/2 + 1 for integer c ≥ 3, based on Ramsey number constructions.