Triangulating Almost-Complete Graphs

Abstract

A triangle decomposition of a graph $G$ is a partition of the edges of $G$ into triangles. Two necessary conditions for $G$ to admit such a decomposition are that $|E(G)|$ is a multiple of three and that the degree of any vertex in $G$ is even; we call such graphs tridivisible. Kirkman's work on Steiner triple systems established that for $G \simeq K_n$, $G$ admits a triangle decomposition precisely when $G$ is tridivisible. In 1970, Nash-Williams conjectured that tridivisiblity is also sufficient for "almost-complete" graphs, which for this talk's purposes we interpret as any graph $G$ on $n$ vertices with $\delta(G) \geq (1 -\epsilon)n, E(G) \geq (1 - \xi)\binom{n}{2}$ for some appropriately small constants $\epsilon, \xi$. Nash-Williams conjectured that $\epsilon = \xi =1/4$ would suffice; in 1991, Gustavsson demonstrated in his dissertation that $\epsilon = \xi < 10^{-24}$ suffices for all $n \equiv 3, 9 \mod 18$, and in 2015 Keevash's work on the existence conjecture for combinatorial designs established that some value of $\epsilon$ existed for any $n$. In this paper, we prove that for any $\epsilon < \frac{1}{432}$, there is a constant $\xi$ such that any $G$ with $\delta(G) \geq (1 - \epsilon)n$ and $|E(G)| \geq (1 - \xi)\binom{n}{2}$ admits such a decomposition, and offer an algorithm that explicitly constructs such a triangulation. Moreover, we note that our algorithm runs in polynomial time on such graphs. (This last observation contrasts with Holyer's result that finding triangle decompositions in general is a NP-complete problem.)