arXiv subjects

Ken-ichi Kawarabayashi

Publications and source records attributed to Ken-ichi Kawarabayashi.

At least 19 recordsLinked to original sources

arXiv2026

The Four Color Theorem with Linearly Many Reducible Configurations and Near-Linear Time Coloring

We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT). Technically speaking, we show the following significant generalization of the 4CT: every planar triangulation contains linearly many pairwise non-touching reducible configurations or pairwise non-crossing obstructing cycles of length at most 5 (which all allow for making effective 4-coloring reductions). The known proofs of the 4CT only show the existence of a single reducible configuration or obstructing cycle in the above statement. The existence is proved using the discharging method based on combinatorial curvature. It identifies reducible configurations in parts where the local neighborhood has positive combinatorial curvature. Our result significantly strengthens the known proofs of 4CT, showing that we can also find reductions in large ``flat" parts where the curvature is zero, and moreover, we can make reductions almost anywhere in a given planar graph. This also opens possibilities for extensions to higher surfaces since we can find such flat parts in any large-width triangulation of any fixed surface. From a computational perspective, the old proofs allowed us to apply induction on a problem that is smaller by some additive constant. The inductive step took linear time, resulting in a quadratic total time. With our linear number of reducible configurations or obstructing cycles, we can reduce the problem size by a constant factor. Our inductive step takes $O(n\log n)$ time, yielding a 4-coloring in $O(n\log n)$ total time. To efficiently handle a linear number of reducible configurations, we need them to be sufficiently robust to be useful in other applications. All our reducible configurations are what is known as D-reducible.

math.CO↗

arXiv2026

An analogue of Reed's conjecture for digraphs

Reed in 1998 conjectured that every graph $G$ satisfies $χ(G) \leq \lceil \frac{Δ(G)+1+ω(G)}{2} \rceil$. As a partial result, he proved the existence of $\varepsilon > 0$ for which every graph $G$ satisfies $χ(G) \leq \lceil (1-\varepsilon)(Δ(G)+1)+\varepsilonω(G) \rceil$. We propose an analogue conjecture for digraphs. Given a digraph $D$, we denote by $\vecχ(D)$ the dichromatic number of $D$, which is the minimum number of colours needed to partition $D$ into acyclic induced subdigraphs. We let $\overleftrightarrowω(D)$ denote the size of the largest biclique (a set of vertices inducing a complete digraph) of $D$ and $\tildeΔ(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We conjecture that every digraph $D$ satisfies $\vecχ(D) \leq \lceil \frac{\tildeΔ(D)+1+\overleftrightarrowω(D)}{2} \rceil$, which if true implies Reed's conjecture. As a partial result, we prove the existence of $\varepsilon >0$ for which every digraph $D$ satisfies $\vecχ(D) \leq \lceil (1-\varepsilon)(\tildeΔ(D)+1)+\varepsilon\overleftrightarrowω(D) \rceil$. This implies both Reed's result and an independent result of Harutyunyan and Mohar for oriented graphs. To obtain this upper bound on $\vecχ$, we prove that every digraph $D$ with $\overleftrightarrowω(D) > \frac{2}{3}(Δ_{\max}(D)+1)$, where $Δ_{\max}(D) = \max_{v\in V(D)} \max(d^+(v),d^-(v))$, admits an acyclic set of vertices intersecting each biclique of $D$, which generalises a result of King. We finally give a short proof that all oriented graphs $D$ satisfy $\vecχ(D) \leq \frac{\sqrt{2}}{2} \tildeΔ(D) + 2$, improving on a result of Golowich.

math.CO↗

arXiv2026

An almost quadratic bound for the minimal excluded minors for a surface

As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs embeddable in a given surface can be characterized by a finite set of minimal excluded minors. However, the proof is purely existential and therefore provides no explicit information about these excluded minors. In 1993, Seymour established the first general upper bound on the order of such minimal excluded minors. Recently, Houdaigoui and Kawarabayashi improved this result by deriving a quasi-polynomial upper bound. Despite this advance, the gap between this bound and the known linear lower bound $Ω(g)$ (where $g$ denotes the genus) remains substantial. In particular, they conjectured that a polynomial upper bound should hold. In this paper, we confirm this conjecture by showing that the order of the minimal excluded minors for a surface of genus $g$ is $g^{2+o(1)}$. This result significantly narrows the gap between the known lower and upper bounds, bringing the asymptotic behavior much closer to the conjectured optimum. Our approach relies on a new structural property of minimal excluded minors. Let $G$ be a minimal excluded minor for a surface of Euler genus $g$. Houdaigoui and Kawarabayashi showed that $G$ contains $O(\log g)$ pairwise disjoint cycles that are contractible and nested in some embedding of $G$. We strengthen this result by proving a separator-based variant: for any contractible subgraph $H \subseteq G$ with a separator of size $s$ (with $H$ contained entirely in one side), the subgraph $H$ contains $O(\log s)$ disjoint cycles that are contractible and nested in some embedding of $G$. This allows us to replace a genus-dependent bound with a separator-dependent one, which is the main new ingredient in deriving our polynomial bound.

math.CO↗

arXiv2026

Three-edge-coloring apex cubic graphs

A graph $G$ is \emph{apex} if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.

math.CO↗

arXiv2026

A directed flat wall theorem excluding a crossrow grid

The graph minor project contains the most influential results in recent undirected graph theory research. There has been progress in recent years in generalising some of their results to directed graphs, with the directed grid theorem of Kawarabayashi and Kreutzer [STOC '15] and the directed flat wall theorem of Giannopoulou, Kawarabayashi, Kreutzer, and Kwon [SODA '22]. We discuss the two different versions of the existing directed flat wall theorem and their drawbacks. Then, we present an alternative directed flat wall theorem that excludes a different digraph as a butterfly minor. This new theorem lies ``in between'' the two existing ones and, as such, does not have either of these drawbacks. The proof of our flat wall theorem is based on the one by Giannopoulou, Kawarabayashi, Kreutzer, and Kwon [SODA '22], which has been adapted by Giannopoulou and Wiederrecht [STOC~'24]. Here we make further adjustments to match our setting.

math.CO↗

arXiv2026

The Balanced Four-Color Theorem

We show that every planar graph with $n \geq 3$ vertices admits a 4-coloring in which each color is used on fewer than $n/2$ vertices. This bound is the best possible. Moreover, such a coloring can be found in $O(n \log n)$ time. We also extend these results to five or more colors and to graphs on general surfaces.

cs.DS↗

arXiv2026

Coloring digraphs with $Δ-b$ colors

The dichromatic number of a digraph is the minimum number of colors needed to partition its vertex set into acyclic subdigraphs. A biclique is a set of vertices inducing all possible pairs of opposite arcs. For a digraph $D$, define $Δ(D) = \max_{v\in V(D)} \sqrt{d^+(v) \cdot d^-(v)}$. We prove that, for every fixed integer $b\in\mathbb{N}$, every digraph $D$ with $Δ(D) = Δ$ being sufficiently large with respect to $b$ either contains a biclique whose size exceeds $Δ-2b$ or has dichromatic number at most $Δ-b$. This extends a classical result of Reed to the directed setting and supports a conjecture of the present authors. Furthermore, the theorem is tight, as for all integers $b$ and $Δ\geq 3b$ there exists a digraph $D$ with $Δ(D)= Δ$, dichromatic number $Δ-b+1$, and whose largest biclique has size $Δ-2b+1$.

math.CO↗

arXiv2026

Connectivities for k-knitted graphs and for minimal counterexamples to Hadwiger's Conjecture

For a given subset $S\subseteq V(G)$ of a graph $G$, the pair $(G,S)$ is \emph{knitted} if for every partition of $S$ into non-empty subsets $S_1, S_2, \ldots, S_t$, there exist pairwise disjoint connected subgraphs $C_1, C_2, \ldots, C_t$ in $G$ such that $S_i\subseteq V(C_i)$ for all $1 \le i \le t$. A graph $G$ is \emph{$\ell$-knitted} if $(G,S)$ is knitted for every subset $S\subseteq V(G)$ of size $\ell$. In this paper, we prove that every $8\ell$-connected graph is $\ell$-knitted. We subsequently apply this result to Hadwiger's Conjecture, which states that every $k$-chromatic graph contains a $K_k$-minor. Specifically, we demonstrate that the vertex connectivity of any minimal counterexample to Hadwiger's Conjecture is at least $\lceil k/8 \rceil$, improving upon the previous lower bound of $\lceil 2k/27 \rceil$ established by Kawarabayashi (2007). Our proof corrects a gap in the argument of Kawarabayashi-Yu~(2013) and establishes the claim stated without proof in Liu--Rolek--Yu~(2019).

math.CO↗

arXiv2026

EPTAS for Hard Graph Cut Problems for Dense Graphs

Everywhere-$δ$-dense graphs are defined as graphs on $n$ vertices in which every vertex has degree at least $δn$ for some constant $δ> 0$. Approximation schemes are vital for handling NP-hard optimization problems, but for many graph cut problems, existing PTAS algorithms often suffer from running times of $n^{f(1/\varepsilon)}$. In this paper, we bring PTASs down to EPTASs for several fundamental minimization problems on everywhere-$Ω(1)$-dense graphs. Specifically, we present the first Efficient Polynomial-Time Approximation Scheme (EPTAS), running in time $f(1/\varepsilon)n^{O(1)}$, for the ConstrainedMinCut problem under a global constraint on vertex weights, a problem that captures BalancedSeparator and SmallSetExpansion. Moreover, we give the first EPTASs for MinQuotientCut and ProductSparsestCut on everywhere-$δ$-dense graphs with integer-valued dense vertex weights; these problems generalize the four well-known problems UniformSparsestCut, EdgeExpansion, Conductance, and NormalizedCut. Our main technical contribution is an EPTAS for ConstrainedMinCut, based on the weak regularity lemma and sampling and estimation techniques. We then obtain EPTASs for MinQuotientCut and ProductSparsestCut via a unified reduction that invokes this algorithm as a subroutine. In contrast, previous works giving PTASs for these problems on everywhere-$δ$-dense graphs typically rely on powerful tools such as the Lasserre hierarchy or specific integer programming technique, which we avoid.

cs.DS↗

arXiv2026

Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width

We prove that every class of Eulerian directed graphs of bounded carving width (equivalently of bounded degree and treewidth) is well-quasi-ordered by strong immersion. In fact, we prove a stronger result, namely that every class of Eulerian directed graphs of bounded carving width, where every vertex is additionally labeled from a well-quasi-order, fixes a linear order on its incident edges, and may impose further restrictions on how the immersion is allowed to route paths through it, is well-quasi-ordered by an adequate notion of strong immersion. To this extent, we develop a framework seemingly suited to prove well-quasi-ordering for classes of Eulerian directed graphs by (strong) immersion and present a first meta theorem in that direction. We complement our results by observing that the class of Eulerian directed graphs of unbounded degree is \emph{not} well-quasi-ordered by \emph{strong} immersion, even if we assume the treewidth of the class to be at most two. We conclude with a dichotomy result, proving for a very restricted class of Eulerian directed graphs of unbounded degree that it is not well-quasi-ordered by strong immersion, but it is well-quasi-ordered by weak immersion.

cs.DM↗

arXiv2026

Edge-colouring and orientations: applications to degree- and $χ$-boundedness

We prove a new generalisation of Ramsey's theorem by showing that every $2$-edge-coloured graph with sufficiently large minimum degree contains a monochromatic induced subgraph whose minimum degree remains large. From this, we also derive that every orientation of a graph with large minimum degree contains either a large transitive tournament or an induced antidirected digraph whose minimum degree is still large. As a consequence, we obtain two general tools showing that certain extensions of degree-bounded graph classes preserve degree-boundedness. A hereditary class $\mathcal{G}$ is {\it degree-bounded} if, for every integer $s$, there exists $d=d(s)$ such that every graph $G\in \mathcal{G}$ either contains $K_{s,s}$ or has minimum degree at most $d$. With these tools, we obtain for instance that odd-signable graphs and Burling graphs are degree-bounded. We also characterise exactly the oriented graphs $F$ such that the graphs admitting an orientation without any induced copy of $F$ are degree-bounded.

math.CO↗

arXiv2026

Online Graph Coloring for $k$-Colorable Graphs

We study the problem of online graph coloring for $k$-colorable graphs. The best previously known deterministic algorithm uses $\widetilde{O}(n^{1-\frac{1}{k!}})$ colors for general $k$ and $\widetilde{O}(n^{5/6})$ colors for $k = 4$, both given by Kierstead in 1998. In this paper, we finally break this barrier, achieving the first major improvement in nearly three decades. Our results are summarized as follows: (1) $k \geq 5$ case. We provide a deterministic online algorithm to color $k$-colorable graphs with $\widetilde{O}(n^{1-\frac{1}{k(k-1)/2}})$ colors, significantly improving the current upper bound of $\widetilde{O}(n^{1-\frac{1}{k!}})$ colors. Our algorithm also matches the best-known bound for $k = 4$ ($\widetilde{O}(n^{5/6})$ colors). (2) $k = 4$ case. We provide a deterministic online algorithm to color $4$-colorable graphs with $\widetilde{O}(n^{14/17})$ colors, improving the current upper bound of $\widetilde{O}(n^{5/6})$ colors. (3) $k = 2$ case. We show that for randomized algorithms, the upper bound is $1.034 \log_2 n + O(1)$ colors and the lower bound is $\frac{91}{96} \log_2 n - O(1)$ colors. This means that we close the gap to a factor of $1.09$. With our algorithm for the $k \geq 5$ case, we also obtain a deterministic online algorithm for graph coloring that achieves a competitive ratio of $O(\frac{n}{\log \log n})$, which improves the best-known result of $O(\frac{n \log \log \log n}{\log \log n})$ by Kierstead. For the bipartite graph case ($k = 2$), the limit of online deterministic algorithms is known: any deterministic algorithm requires $2 \log_2 n - O(1)$ colors. Our results imply that randomized algorithms can perform slightly better but still have a limit.

cs.DS↗

arXiv2026

Accelerating Scientific Research with Gemini: Case Studies and Common Techniques

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

cs.CL↗

arXiv2025

A quasi-polynomial bound for the minimal excluded minors for a surface

As part of their graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because existential, does not provide any information on these excluded minors. Seymour proved in 1993 the first and, until now, only known upper bound on the order of the minimal excluded minors for a given surface. This bound is double exponential in the Euler genus $g$ of the surface and, therefore, very far from the $Ω(g)$ lower bound on the maximal order of minimal excluded minors for a surface and most likely far from the best possible bound. More than thirty years later, this paper finally makes progress in lowering this bound to a quasi-polynomial in the Euler genus of the surface. The main catalyzer to reach a quasi-polynomial bound is a breakthrough on the characteristic size of a forbidden structure for a minimal excluded minor $G$ for a surface of Euler genus $g$: although it is not hard to show that $G$ does not contain $O(g)$ disjoint cycles that are contractible and nested in some embedding of $G$ as demonstrated by Seymour, this bound can be lowered to $O(\log g)$ which is essential to obtain the quasi-polynomial bound in this paper. As subsidiary results, we also improve the current bound on the treewidth of a minimal excluded minor $G$ for a surface by improving the first and, until now, only known bound provided by Seymour.

math.CO↗

arXiv2025

5-Coloring Planar Graphs with a Color Class of Order at Most $|V|/6$

We show that any planar graph $G=(V,E)$ has a 5-coloring such that one color class contains at most $|V|/6$ vertices. In other words, there exists a partition of $V$ into five independent sets $\{V_1, \cdots, V_5\}$ such that $|V_5| \leq |V| / 6$. Our proof yields an $O(|V|^2)$-time algorithm to find such a partition, and unlike the Four Color Theorem, our proof is fully verifiable without computer assistance.

math.CO↗

arXiv2025

Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion

We prove that for every surface $Σ$, the class of Eulerian directed graphs that are Eulerian embeddable into $Σ$ (in particular they have degree at most $4$) is well-quasi-ordered by strong immersion. This result marks one of the most versatile directed graph classes (besides tournaments) for which we are aware of a positive well-quasi-ordering result regarding a well-studied graph relation. Our result implies that the class of bipartite circle graphs is well-quasi-ordered under the pivot-minor relation. Furthermore, this also yields two other interesting applications, namely, a polynomial-time algorithm for testing immersion closed properties of Eulerian-embeddable graphs into a fixed surface, and a characterisation of the Erdős-Pósa property for Eulerian digraphs of maximum degree four. Further, in order to prove the mentioned result, we prove that Eulerian digraphs of carving width bounded by some constant $k$ (which correspond to Eulerian digraphs with bounded treewidth and additionally bounded degree) are well-quasi-ordered by strong immersion. We actually prove a stronger result where we allow for vertices of the Eulerian digraphs to be labeled by elements of some well-quasi-order $Ω$. We complement these results with a proof that the class of Eulerian planar digraphs of treewidth at most $3$ is not well-quasi-ordered by strong immersion, noting that any antichain of bounded treewidth cannot have bounded degree.

cs.DM↗

arXiv2025

The Directed Disjoint Paths Problem with Congestion

The classic result by Fortune, Hopcroft, and Wyllie [TCS~'80] states that the directed disjoint paths problem is NP-complete even for two pairs of terminals. Extending this well-known result, we show that the directed disjoint paths problem is NP-complete for any constant congestion $c \geq 1$ and~$k \geq 3c-1$ pairs of terminals. This refutes a conjecture by Giannopoulou et al. [SODA~'22], which says that the directed disjoint paths problem with congestion two is polynomial-time solvable for any constant number $k$ of terminal pairs. We then consider the cases that are not covered by this hardness result. The first nontrivial case is $c=2$ and $k = 3$. Our second main result is to show that this case is polynomial-time solvable.

cs.DM↗

arXiv2025

Directed disjoint paths remains W[1]-hard on acyclic digraphs without large grid minors

In the Vertex Disjoint Paths with Congestion problem, the input consists of a digraph $D$, an integer $c$ and $k$ pairs of vertices $(s_i, t_i)$, and the task is to find a set of paths connecting each $s_i$ to its corresponding $t_i$, whereas each vertex of $D$ appears in at most $c$ many paths. The case where $c = 1$ is known to be NP-complete even if $k = 2$ [Fortune, Hopcroft and Wyllie, 1980] on general digraphs and is W[1]-hard with respect to $k$ (excluding the possibility of an $f(k)n^{O(1)}$-time algorithm under standard assumptions) on acyclic digraphs [Slivkins, 2010]. The proof of [Slivkins, 2010] can also be adapted to show W[1]-hardness with respect to $k$ for every congestion $c \geq 1$. We strengthen the existing hardness result by showing that the problem remains W[1]-hard for every congestion $c \geq 1$ even if: - the input digraph $D$ is acyclic, - $D$ does not contain an acyclic $(5, 5)$-grid as a butterfly minor, - $D$ does not contain an acyclic tournament on 9 vertices as a butterfly minor, and - $D$ has ear-anonymity at most 5. Further, we also show that the edge-congestion variant of the problem remains W[1]-hard for every congestion $c \geq 1$ even if: - the input digraph $D$ is acyclic, - $D$ has maximum undirected degree 3, - $D$ does not contain an acyclic $(7, 7)$-wall as a weak immersion and - $D$ has ear-anonymity at most 5.

cs.CC↗