On fractional triangle decompositions of random graphs

TL;DR

This work proves that the random graph $G(n,p)$ admits a fractional triangle decomposition (FTD) w.h.p. for $p \ge n^{-4/11+\varepsilon}$, improving the previous $n^{-1/3+\varepsilon}$ bound. The authors present an algorithmic construction: start with a uniform triangle weighting to obtain an approximate FTD, then iteratively refine via local gadgets—bowtie structures to balance vertex demands and octagonal pinwheels to redistribute edge discrepancies—to converge to a genuine FTD. The analysis hinges on precise concentration results (Chernoff bounds and Kim–Vu polynomial concentration) to control the counts and interactions of gadgets in $G(n,p)$, along with a detailed combinatorial property framework that holds w.h.p. in the prescribed density regime. The approach is designed to be robust and extensible, with conjectures about the true threshold near $p \approx \sqrt{\frac{3\log n}{2n}}$ and potential generalization to fractional $K_k$-decompositions, highlighting a pathway toward resolving fundamental questions about fractional decompositions in random graphs.

Abstract

We prove that with high probability $G(n,p)$ with $p \geq n^{-4/11 + o(1)}$ admits a fractional triangle decomposition (FTD), i.e., a nonnegative weighting of its triangles such that for each edge, the total weight of the triangles containing it equals one. This improves on the state of the art, due to Delcourt, Kelly, and Postle, that $p \geq n^{-1/3+o(1)}$ suffices. The proof is algorithmic: Given $G \sim G(n,p)$, we first construct an approximate FTD by taking a uniform weighting of the triangles. We then use specialized gadgets to iteratively shift weights and obtain successively better approximations of an FTD.

Figures (9)

• Figure 1: A $(u,v)$-bowtie. The shaded triangles are given weight $1$ while the unshaded triangles are given weight $-1$.
• Figure 2: A $(u,v)$-octagonal pinwheel. The shaded triangles are given weight $1$, while the unshaded triangles are given weight $-1$.
• Figure 3: The graph $W(3)$.
• Figure 4: The graph $W(2)$, corresponding to the case $(i,j) = (2,7)$. If the red vertices $\{ d,a_7,b_6,b_7 \}$ are fixed, the rooted extension to the gray vertices $\{ c_a,a_0,a_1,c_b \}$ has $11$ edges and four vertices, giving density $11/4$.
• Figure 5: The graph $W(1,2)$ and the case $(1,2,6)$. The vertex set $\{d,a_7,\vartheta_5,\vartheta_{6}\}$ is highlighted in red.

Theorems & Definitions (43)

• Theorem 1.1
• Conjecture 1.2
• Definition 2.1
• Theorem 4.1
• Theorem 4.2: alon_spencer_probabilistic_method_3rd
• Corollary 4.3
• proof
• Definition 4.4
• Claim 4.5
• proof