Playing Sudoku on random 3-regular graphs

TL;DR

This work bounds the Sudoku number for the random cubic graph $\mathcal{G}_{n,3}$ by constructing a Sudoku set of size at most $(1+o(1))\frac{n}{3}$ a.a.s. via an algorithm that 3-colours the graph while generating many locally forced vertices; the analysis combines a specialized differential equations method with a Markov-chain approximation of local dynamics. The authors work within the configuration model, leveraging contiguity with the random graph $G_n$ (the union of a random Hamilton cycle and a random perfect matching) to transfer results to $\mathcal{G}_{n,3}$. They introduce a detailed classification of vertex types and a 18-state Markov chain to model local transitions, and they prove a burn-in plus completion strategy to control the main contribution from Sudoku vertices to be about $n/3$. The paper also conjectures the true asymptotic Sudoku number for $\mathcal{G}_{n,3}$ is $(1+o(1))\frac{n}{4}$, aligning with deterministic lower bounds, and outlines directions for extending the approach to $d$-regular graphs with $d\ge 4$.

Abstract

The Sudoku number $s(G)$ of graph $G$ with chromatic number $χ(G)$ is the smallest partial $χ(G)$-colouring of $G$ that determines a unique $χ(G)$-colouring of the entire graph. We show that the Sudoku number of the random $3$-regular graph $\mathcal{G}_{n,3}$ satisfies $s(\mathcal{G}_{n,3}) \leq (1+o(1))\frac{n}{3}$ asymptotically almost surely. We prove this by analyzing an algorithm which $3$-colours $\mathcal{G}_{n,3}$ in a way that produces many locally forced vertices, i.e., vertices which see two distinct colours among their neighbours. The intricacies of the algorithm present some challenges for the analysis, and to overcome these we use a non-standard application of Wormald's differential equations method that incorporates tools from finite Markov chains.

Figures (2)

• Figure 1: The left graph illustrates a Sudoku set of a random $3$-regular graph with $70$ vertices, generated using a Hamilton cycle and a random perfect matching. The Sudoku set shown is of minimum size for this graph. The right graph shows the proper colouring of the whole graph determined by the Sudoku set.
• Figure 2: The types. Here we use blue for colour $k$, red for $m$, and green for $\ell$. The pointer is indicated by an upward arrow. Note that if vertex $i$ is of type $B_{f}^{(km)}$, we must have $\textbf{ptr}(i) = i-1$ as indicated in the right-most figure.

Theorems & Definitions (34)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3: mahmoodian1999defining
• Theorem 1.4: wormald2001decycling
• Proposition 1.5: Chernoff bounds with negative correlation, Doerr2020
• Proposition 1.6: Azuma's inequality, wormald1999differential
• Lemma 2.1
• proof
• Lemma 2.2
• proof