Asymptotics for Palette Sparsification from Variable Lists

TL;DR

The work proves a sharp palette-sparsification result for variable lists in graph coloring: for every ε>0, a graph G with maximum degree D and vertex-wise lists S_v of size D+1 admits an L-coloring with σ_v ∈ L_v when each L_v is chosen as a random (1+ε)log n-subset of S_v, with probability tending to 1 as D→∞. The method combines a two-phase graph decomposition into a sparse part and dense clusters, a two-stage randomized construction to obtain a partial coloring with favorable slack (via T and σ), and a dense-phase matching analysis to extend the coloring cluster-by-cluster. Central to the argument are martingale concentration tools (aglc), Local Lemma-based arguments, and a suite of matching lemmas (e.g., LMg'A) that guarantee the existence of perfect matchings in the restricted random bipartite graphs that arise in the dense phase. The results extend the palette sparsification program of Ack, Chen, Khanna and collaborators to variable lists, with implications for algorithmic coloring in sparse graphs and the robustness of sparsified palettes in random-list models.

Abstract

It is shown that the following holds for each $\varepsilon >0$. For $G$ an $n$-vertex graph of maximum degree $D$, lists $S_v$ of size $D+1$ (for $v\in V(G)$), and $L_v$ chosen uniformly from the ($(1+\varepsilon)\ln n$)-subsets of $S_v$ (independent of other choices), \[ \mbox{$G$ admits a proper coloring $σ$ with $σ_v\in L_v$ $\forall v$} \] with probability tending to 1 as $D\to \infty$. When each $S_v $ is $\{1\dots D+1\}$, this is an asymptotically optimal version of the ``palette sparsification'' theorem of Assadi, Chen and Khanna that was proved in an earlier paper by the present authors.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Proposition 2.1
• Lemma 2.2
• Theorem 2.3
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3