arXiv subjects

Yuta Inoue

Publications and source records attributed to Yuta Inoue.

8 recordsLinked to original sources

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

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↗

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

Three-edge-coloring (Tait coloring) cubic graphs on the torus: A proof of Gr\"unbaum's conjecture

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exceptional graph class called "Petersen-like", is 3-edge-colorable. This means every (non-trivial) toroidal snark can be obtained from several copies of the Petersen graph using the dot product operation. The first two snarks in this family are the Petersen graph and one of Blanu\v{s}a snarks; the rest are exposed by Vodopivec in 2008. This proves a strengthening of the well-known, long-standing conjecture of Gr\"unbaum from 1968. This implies that a 2-connected cubic (multi)graph that can be embedded in the torus is not 3-edge-colorable if and only if it can be obtained from a dot product of copies of the Petersen graph by replacing its vertices with 2-edge-connected planar cubic (multi)graphs. Here, replacing a vertex $v$ in a cubic graph $G$ is the operation that takes a 2-connected planar cubic multigraph $H$ and one of its vertices $u$ of degree 3, unifying $G-v$ and $H-u$ and connecting the neighbors of $v$ in $G-v$ with the neighbors of $u$ in $H-u$ with a matching. This result is a highly nontrivial generalization of the Four Color Theorem, and its proof requires a combination of extensive computer verification and computer-free extension of existing proofs on colorability. An important consequence of this result is a very strong version of the Tutte 4-Flow Conjecture for toroidal graphs. We show that a 2-edge connected graph embedded in the torus admits a nowhere-zero 4-flow unless it is Petersen-like (in which case it does not admit nowhere-zero 4-flows). Observe that this is a vast strengthening over the Tutte 4-Flow Conjecture on the torus, which assumes that the graph does not contain the Petersen graph as a minor because almost all toroidal graphs contain the Petersen graph minor, but almost none are Petersen-like.

math.CO↗

arXiv2025

Scalable and Performant Data Loading

We present SPDL (Scalable and Performant Data Loading), an open-source, framework-agnostic library designed for efficiently loading array data to GPU. Data loading is often a bottleneck in AI applications, and is challenging to optimize because it requires coordination of network calls, CPU-bound tasks, and GPU device transfer. On top of that, Python's GIL (Global Interpreter Lock) makes it difficult to gain performance improvement from multi-threading. We found that when data preprocessing functions release the GIL entirely, it is possible to execute them concurrently in a thread pool, thereby improving the workflow performance. Our benchmark shows that compared to the PyTorch DataLoader, SPDL can iterate through the ImageNet dataset 74% faster while using 38% less CPU and 50GB less memory. When training ViT-B/16 model, SPDL can send data to the GPU at a speed that does not starve the training. Additionally, when using SPDL on Python 3.13t, without changing any code, the throughput is further by improved by 33%, thanks to the disabled GIL. SPDL can improve the performance of current AI model training, and receives further performance improvements when Free-Threaded Python is adopted in production systems. SPDL is available at https://github.com/facebookresearch/spdl.

cs.DC↗

arXiv2024

Three-edge-coloring projective planar cubic graphs: A generalization of the Four Color Theorem

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the projective plane, with the single exception of the Petersen graph, is 3-edge-colorable. In other words, the only (non-trivial) snark that can be embedded in the projective plane is the Petersen graph. This implies that a 2-connected cubic (multi)graph that can be embedded in the projective plane is not 3-edge-colorable if and only if it can be obtained from the Petersen graph by replacing each vertex by a 2-edge-connected planar cubic (multi)graph. This result is a nontrivial generalization of the Four Color Theorem, and its proof requires a combination of extensive computer verification and computer-free extension of existing proofs on colorability. An unexpected consequence of this result is a coloring-flow duality statement for the projective plane: A cubic graph embedded in the projective plane is 3-edge-colorable if and only if its dual multigraph is 5-vertex-colorable. Moreover, we show that a 2-edge connected graph embedded in the projective plane admits a nowhere-zero 4-flow unless it is Peteren-like (in which case it does not admit nowhere-zero 4-flows). This proves a strengthening of the Tutte 4-flow conjecture for graphs on the projective plane. Some of our proofs require extensive computer verification. The necessary source codes, together with the input and output files and the complete set of more than 6000 reducible configurations are available on Github (https://github.com/edge-coloring) which can be considered as an Addendum to this paper. Moreover, we provide pseudocodes for all our computer verifications.

math.CO↗

arXiv2020

Phonon structure of titanium under shear deformation along $\{10\bar{1}2\}$ twinning mode

We investigated phonon behavior of hexagonal close packed titanium under homogeneous shear deformation corresponding to the $\{10\bar{1}2\}$ twinning mode using first-principles calculation and phonon calculation. By this deformation, we found that a phonon mode located at a point on Brillouin zone boundary is drastically soften increasing the shear and finally it triggers a spontaneous structural transition by breaking the crystal symmetry toward twin from parent.

cond-mat.mtrl-sci↗

arXiv2011

Defaultable Bonds via HKA

To construct a no-arbitrage defaultable bond market, we work on the state price density framework. Using the heat kernel approach (HKA for short) with the killing of a Markov process, we construct a single defaultable bond market that enables an explicit expression of a defaultable bond and credit spread under quadratic Gaussian settings. Some simulation results show that the model is not only tractable but realistic.

q-fin.CP↗