Vizing's Theorem in Near-Linear Time

TL;DR

The paper tackles the classic problem of edge coloring graphs with Δ+1 colors, achieving a near-linear-time randomized algorithm. It introduces a novel framework built around color-type reduction, u-fans, Vizing fans, and separable collections, enabling efficient color-extension on the final uncolored edges. The main contributions are near-linear runtimes Ō(m) (with polylog factors) for Δ+1-edge coloring, along with variants Ō(m log n) and Ō(m log Δ) under different regimes, and extensions to multigraphs and Shannon-type results. This work significantly advances the practicality of Vizing-type edge-coloring algorithms, offering scalable performance in dense graphs and broad applicability to related coloring problems.

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, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a $(Δ+1)$-edge coloring in near-linear time -- in fact, only $O(m\logΔ)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem.

Figures (8)

• Figure 1: In this picture, we attempt to popularize edge $(u, v)$ by flipping the $\{\alpha, \gamma\}$-alternating path from $u$ and the $\{ \beta, \lambda\}$-alternating path from $v$. However, flipping the $\{\alpha, \gamma \}$-alternating path from $u$ makes a previously popular edge $(u', v')$ unpopular as $u'$ will not miss color $\alpha$ anymore.
• Figure 2: In this picture, $(u', v')\in \Phi$ with $c_{u'} = \alpha, c_{v'} = \beta$. For each uncolored edge $(u_i, v_i)$, flipping the $\{c_i, \beta\}$-alternating path from $v_i$ would damage the property that $c_{v'} = \beta$. Fortunately, there are at most $\Delta$ many different such $(u_i, v_i)$ because each of them is at the end of an $\{\beta, \cdot\}$-alternating path starting at $v'$.
• Figure 3: In this picture, we have two different uncolored edges $(u_1, v_1), (u_2, v_2)$ such that $\alpha\in \mathsf{miss}_{\chi}(u_1)\cap \mathsf{miss}_{\chi}(u_2)$, and their Vizing chains first intersect at edge $(x,y)$ which currently has color $\alpha$ under $\chi$. Then we can rotate their Vizing fans and flip part of their Vizing chains to shift $(u_1, v_1), (u_2, v_2)$ to $(w_1, x), (w_2, x)$ respectively to form a u-fan; note that $\alpha\in \mathsf{miss}_{\chi}(w_1)\cap \mathsf{miss}_{\chi}(w_2)$ after this shifting procedure.
• Figure 4: This picture shows two u-fans $(u_1, x_1, x, *, \beta_1)$ and $(u_2, x_2, x, *, \beta_2)$ sharing a common vertex $x$. The separable condition requires that $\beta_1\neq \beta_2$; for instance $\beta_1, \beta_2$ could be magenta and cyan as shown here.
• Figure 5: In this picture, we look at the Vizing fan around an uncolored edge $(u_1, v_1)$, and it intersects with the Vizing fan of another edge $(u_2, v_2)$ currently residing in $\mathcal{F}$ at vertex $w$. Then, we can rotate both Vizing fans and pair these two uncolored edges as a u-fan.

Theorems & Definitions (138)

• Theorem 1.1
• Theorem 3.1
• Lemma 3.2
• proof : Proof of \ref{['thm:main:bipartite']}
• Claim 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof