2-Coloring Cycles in One Round

Abstract

We show that there is a one-round randomized distributed algorithm that can 2-color cycles such that the expected fraction of monochromatic edges is less than 0.24118. We also show that a one-round algorithm cannot achieve a fraction less than 0.23879. Before this work, the best upper and lower bounds were 0.25 and 0.2. Our proof was largely discovered and developed by large language models, and both the upper and lower bounds have been formalized in Lean 4.

Figures (3)

• Figure 1: The normal De Bruijn graph $\mathop{\mathrm{DB}}\nolimits_{\mathop{\mathrm{normal}}\nolimits}(n)$ for $n=2$, together with a $2$-coloring (blue--orange) where only $4$ out of $16$ edges are monochromatic. It is also easy to see that there is no $2$-coloring with fewer monochromatic edges (consider two self-loops and two triangles). Hence $p^* \leqslant p_{\mathop{\mathrm{normal}}\nolimits}(2) = 1/4$.
• Figure 2: Selected parts of the distinct De Bruijn graph $\mathop{\mathrm{DB}}\nolimits_{\mathop{\mathrm{distinct}}\nolimits}(n)$ for $n=5$. The graph has a total of $60$ nodes and $120$ edges. The edges can be partitioned into $24$ cycles of length $5$: for each edge $\bigl((a,b,c),(b,c,d)\bigr)$ there is a unique element $e \in [5] \setminus \{a,b,c,d\}$, and we assign this edge to the $5$-cycle formed by $(a,b,c)$, $(b,c,d)$, $(c,d,e)$, $(d,e,a)$, and $(e,a,b)$. In each such $5$-cycle, at least one edge is monochromatic; therefore, there is no $2$-coloring with fewer than $24$ monochromatic edges. On the other hand, exactly $24$ monochromatic edges can be achieved, e.g., by coloring nodes with labels $(0,y,z)$ and $(x,y,0)$ orange and all other nodes blue, as shown in the figure. Hence $p^* \geqslant p_{\mathop{\mathrm{distinct}}\nolimits}(5) = 1/5$.
• Figure 3: Illustrations of algorithms.

Theorems & Definitions (12)

• Remark
• Remark
• Remark
• Theorem 1
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• proof