An algorithmic Vizing's theorem: toward efficient edge-coloring sampling with an optimal number of colors

TL;DR

This work addresses the problem of efficiently sampling proper edge-colorings using the optimal number of colors $k\ge Δ+1$ (algorithmic Vizing's theorem). It employs a randomized hill-climbing approach that starts from an arbitrary $k$-edge-coloring and recolors a single edge only if the potential $\psi(\sigma)=\sum_{v}\sum_{\alpha}{d_{\sigma,\alpha}(v)\choose 2}$ does not increase, providing an algorithmic interpretation of Vizing's theorem. The main result proves that for any graph with maximum degree $Δ$, the algorithm almost surely reaches a proper $k$-edge-coloring in polynomial time, via a structured progression through monochromatic components, cherry colorings, and a color-shift digraph which guides monotone recolorings. The paper also discusses conjectures on efficiency and uniformity of the sampled colorings and situates the work within broader questions about sampling under constrained color counts, including implications for Class I graphs and connections to combinatorial designs.

Abstract

The problem of sampling edge-colorings of graphs with maximum degree $Δ$ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to $Δ$. Vizing's theorem guarantees the existence of a $(Δ+1)$-edge-coloring, raising the natural question of how to efficiently sample such edge-colorings. In this paper, we take an initial step toward addressing this question. Building on the approach of Dotan, Linial, and Peled, we analyze a randomized algorithm for generating random proper $(Δ+1)$-edge-colorings, which in particular provides an algorithmic interpretation of Vizing's theorem. The idea is to start from an arbitrary non-proper edge-coloring with the desired number of colors and at each step, recolor one edge uniformly at random provided it does not increase the number of conflicting edges (a potential function will count the number of pairs of adjacent edges of the same color). We show that the algorithm almost surely produces a proper $(Δ+1)$-edge-coloring and propose several conjectures regarding its efficiency and the uniformity of the sampled colorings.

Figures (7)

• Figure 1: A frozen $(2\Delta - 1)$-edge-coloring of a Kneser graph
• Figure 2: Potential on a $4$-star for some colorings.
• Figure 3: On the left, the color $\alpha$ appears at least four times around $u$ and $v$ (counting $uv$ twice). By \ref{['lem:monosize2']}, we can reduce the potential by recoloring $uv$ into a color appearing at most once around $u$ and $v$.
• Figure 4: Illustration of \ref{['prop:movecherry']}. We monotonically recolor the edge $uv$ of the $\alpha$-cherry (centered at $u$ here).
• Figure 5: Illustration of \ref{['lem:bichromaticcycle']}. A $\gamma_1$-cherry $c$ (orange here) whose center $v$ misses the color $\gamma_2$ (violet here). The $(\gamma_1, \gamma_2)$-component $H$ containing $c$ has a vertex $w$ of degree one. First, we apply \ref{['lem:exchangecherry']} to construct in $H$ a path $P$ of vertices of degree at most $2$ from $v$ to a vertex $x$ of degree $1$. Then, we apply \ref{['cl:pathbi']} on $P$ to "move" the center of the cherry from $v$ to $x_2$. Finally, we can just recolor $x_1x_2$ to decrease the potential.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof