Vizing's Theorem in Deterministic Almost-Linear Time

TL;DR

This work resolves a long-standing question on the deterministic complexity of edge coloring with Δ+1 colors by presenting an almost-linear time algorithm. The authors replace sublinear-time color-reduction subroutines with a novel color-type sparsification framework, enabling deterministic progress on a large fraction of edges without randomization. The approach relies on u-fans, separable collections, and carefully maintained alternating-path flips, organized via three subroutines: Sparsify-Types, Color-Small, and Extend-Coloring. The result achieves a deterministic runtime of $m \cdot 2^{O(\sqrt{\log Δ})} \cdot \log n = m^{1+o(1)}$, breaking the prior $\tilde{O}(m\sqrt{n})$ barrier and advancing the practical understanding of near-linear deterministic edge coloring.

Abstract

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors. Vizing's original proof is easily translated into a deterministic $O(mn)$ time algorithm. This deterministic time bound was subsequently improved to $\tilde O(m \sqrt n)$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. A series of recent papers improved the time bound of $\tilde O(m\sqrt{n})$ using randomization, culminating in the randomized near-linear time $(Δ+1)$-coloring algorithm by [Assadi, Behnezhad, Bhattacharya, Costa, Solomon, and Zhang, 2025]. At the heart of all of these recent improvements, there is some form of a sublinear time algorithm. Unfortunately, sublinear time algorithms as a whole almost always require randomization. This raises a natural question: can the deterministic time complexity of the problem be reduced below the $\tilde O(m\sqrt{n})$ barrier? In this paper, we answer this question in the affirmative. We present a deterministic almost-linear time $(Δ+1)$-coloring algorithm, namely, an algorithm running in $m \cdot 2^{O(\sqrt{\log Δ})} \cdot \log n = m^{1+o(1)}$ time. Our main technical contribution is to entirely forego sublinear time algorithms. We do so by presenting a new deterministic color-type sparsification approach that runs in almost-linear (instead of sublinear) time, but can be used to color a much larger set of edges.

Figures (9)

• Figure 1: In this example, we have $\eta = 10$. The matrix heatmap is initially the left one. After we run the algorithm of \ref{['th:bipartite:sparsification']}, most weights are concentrated along the diagonal.
• Figure 2: The mapping $\phi_{i \to j}$ defines a matching between colors in $C_i, C_j$. In this example we have drawn $5$ colors from color sets $C_i, C_j$, and colors connected by strings are matched together.
• Figure 3: In this example, we attempt to align edge $(u, v)$ by flipping the $\{\alpha, \alpha_{2k-1}\}$-alternating path $P_u(e)$ from $u$ and the $\{ \beta, \beta_{2k}\}$-alternating path $P_v(e)$ from $v$. However, flipping the $\{\alpha, \alpha_{2k-1} \}$-alternating path from $u$ damages a previously aligned edge $(u', v')$ as $u'$ will not miss color $\alpha_{2k-1}$ anymore. In this case, $(u', v')$ is responsible for $(u, v)$.
• Figure 4: In this picture, $e' = (u', v') \in U_{k'}$, for some $k'<k$. We have drawn two alternating paths of types $\{\alpha', \phi_{(2k'-1) \to i}(\alpha')\}$ and $\{\beta', \phi_{(2k') \to i}(\beta')\}$ from $u', v'$ which could damage some edges $e^\star$.
• Figure 5: In this picture, $e' = (u', v') \in U_{k}$. We have drawn some alternating paths of types $\{\alpha', \phi_{(2k'-1) \to i}(\alpha')\}$ and $\{\beta', \phi_{(2k') \to i}(\beta')\}$ from $u', v'$ for three choices of $i\in [2\eta]\setminus \{2k-1, 2k\}$.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.3
• proof
• proof
• proof
• Claim 2.8
• proof
• Corollary 2.9
• proof