An $O(n^{0.1537})$-Coloring Algorithm for 3-Colorable Graphs

Kaisei Deguchi; Ken-ichi Kawarabayashi; Hirotaka Yoneda; Masataka Yoneda

Home/Archive/arXiv

arXiv · 2610.35859

An $O(n^{0.1537})$-Coloring Algorithm for 3-Colorable Graphs

Kaisei Deguchi·Ken-ichi Kawarabayashi·Hirotaka Yoneda·Masataka Yoneda

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a polynomial-time randomized algorithm that colors every $n$-vertex $3$-colorable graph with $O(n^{0.1537})$ colors. This improves upon the previous bound of $O(n^{0.1954})$ colors by Bansal, Huang, and Lee (SODA '27). To achieve this result, we introduce a new algorithmic framework for finding a large independent set from a solution to the vector coloring SDP. Combined with the (combinatorial) dense-progress algorithm of Kawarabayashi, Thorup, and Yoneda (STOC '24), this framework yields an $n^{1/6 + o(1)}$-coloring algorithm. Furthermore, we strengthen the SDP-based algorithm specifically for 3-colorable graphs to achieve $O(n^{0.1537})$ colors.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Kaisei Deguchi, Ken-ichi Kawarabayashi, Hirotaka Yoneda, Masataka Yoneda. 2026-10-26. An $O(n^{0.1537})$-Coloring Algorithm for 3-Colorable Graphs. https://arxiv.org/abs/2610.35859

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Online Multi-Level Aggregation with Maximum Delay

We study Multi-Level Aggregation with Maximum Delay (MLA-MD) on finite weighted rooted trees. Each service pays for a rooted subtree and for the maximum waiting time among the requests it clears. For the offline problem, we give an exact polynomial-time algorithm that computes both the optimum and an optimal service schedule. For the online problem, we give a deterministic $2$-competitive algorithm and, against an oblivious adversary, a randomized $e/(e-1)$-competitive algorithm. Both competitive ratios are optimal, even on rooted paths. We also extend the offline result and both online guarantees to service systems with submodular costs.

cs.DS↗

arXiv2026

Truly Work-efficient Parallel Deterministic $(Δ+1)$-coloring and Maximal Independent Set

We give deterministic parallel algorithms that compute a $(Δ+1)$-coloring and a maximal independent set for a simple graph with $n$ vertices and $m$ edges in $O(n+m)$ work and $O(\mathrm{poly}\log n)$ depth.

cs.DS↗

arXiv2026

Exact algorithms for optimal discretization

The optimal discretization problem asks, given two disjoint sets of points $R$ and $B$ in the plane, for a minimal family of horizontal and vertical lines that separate the two sets, so that no cell delimited by the lines contains points from both sets. The problem arises as a pre-processing in supervised machine learning, and has received significant attention in parameterized algorithmics. Answering the question raised by Bonnet, Giannopoulos, and Lampis [IPEC 2017] and Froese [PhD thesis, 2018], it was shown by Kratsch, Masařík, Muzi, Pilipczuk, and Sorge [SODA 2021] that optimal discretization admits a fixed-parameter algorithm with running time $2^{\mathcal{O}(k^2 \log k)} \cdot n^{\mathcal{O}(1)}$, where $k$ is the solution size and $n = |R| + |B|$. In this paper we give an algorithm for optimal discretization that runs in time $\mathcal{O}(1.9602^n)$. We also study the related point separation problem that asks to separate all input points by axis-parallel lines. For this problem we obtain an algorithm with runtime $\mathcal{O}(1.8899^n)$. Our guarantees follow from structural observations about bichromatic and monochromatic point sets. To our knowledge, these are the first improvements over the trivial $2^n$ bound for both problems.

cs.DS↗