We show that for all $k$, there exists $\varepsilon_k>0$ such that graph $k$-coloring can be solved by a randomized algorithm with one-sided error in time $O((2-\varepsilon_k)^n)$. Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the $2^n \cdot \mathrm{poly}(n)$-time algorithm of Bj\"orklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for $k \le 6$.
This work generalizes and combines tools from the $(k+2)-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023] and yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes.
The main result shows that the exact average-case complexity of this fundamental problem is $\Theta(nk)$ for every $k \leq n^{c'}$ and some $c'\in (0, 1)$, and reveals the average sublinear nature of $k$-colorability: the average-case complexity is linear in $n$, and thus sublinear in the size of the input.
Cassandra Marcussen, Edward Pyne, R. Rubinfeld et al.· 0 citations
We study the parameterized complexity of $k$-Coloring in $H$-free graphs, when $H$ is a linear forest (i.e., a disjoint union of paths) as an induced subgraph. We show two hardness results: * $k$-Coloring is W[1]-hard in $2P_2$-free graphs when parameterized by $k$. * $3$-Coloring is W[1]-hard in $P_t$-free graphs when parameterized by $t$. Moreover, assuming the ETH, these problems admit no algorithms solving $n$-vertex instances in time $f(k) \cdot n^{o(k)}$ and $f(t) \cdot n^{o(t/\log t)}$, respectively, for any computable function $f$. The first result resolves in a strong form a long-standing open problem, originally posed by Ho\`ang, Kami\'nski, Lozin, Sawada, and Shu [Algorithmica, 2010]. The second result answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017].
The van der Waerden number $w(k)$ is the smallest positive integer $N$ such that every two-coloring of $\{1,2,\ldots,N\}$ contains a monochromatic $k$-term arithmetic progression. We prove that $w(k) \geq (1-o(1))k2^{k-1}$ holds for all positive integers $k$. This verifies a conjecture of Erd\H{o}s. In 1968, Berlekamp proved the same result when $k-1$ is prime. The coloring for general $k$ can be viewed as a product of Berlekamp's colorings for various primes. It was found by ChatGPT 5.6 Sol Pro.
Marcelo Campos, Jacob Fox, Carl Schildkraut· 0 citations
Nguyen~\cite{Nguyen2025} recently proved that every $\{P_5,C_5\}$-free graph $G$ satisfies $\chi(G)\leq \omega(G)^{40}$. Building on his framework, we introduce two refinements, namely a sharper cutset decomposition using the $C_5$-free condition and an improved density-increment argument. These yield a polynomial $\chi$-binding function with exponent $24$, improving the previous bound of $40$.
A graph $G$ is said to be $k$-collapsible if $G$ has minimum degree $k$ and every non-null proper induced subgraph of $G$ has minimum degree less than $k.$ In 2018, Bickle conjectured that the minimum number of vertices of degree $k$ in a $k$-collapsible graph of order $n$ with $k\ge 3$ is ${\rm max}\{\lceil 2n/(2k-1)\rceil,\, k^2-k-2-(k-3)n\}.$ We prove this conjecture.