Small Minors in Dense Graphs

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe functions $f$ and $h$ such that every graph with $n$ vertices and average degree at least $f(t)$ contains a $K_t$-model with at most $h(t)\cdot\log n$ vertices. The logarithmic dependence on $n$ is best possible (for fixed $t$). In general, we prove that $f(t)\leq 2^{t-1}+\eps$. For $t\leq 4$, we determine the least value of $f(t)$; in particular $f(3)=2+\eps$ and $f(4)=4+\eps$. For $t\leq4$, we establish similar results for graphs embedded on surfaces, where the size of the $K_t$-model is bounded (for fixed $t$).

💡 Research Summary

The paper revisits the classical theorem that a graph with sufficiently large average degree must contain a large complete graph as a minor, and strengthens it by controlling the size of the minor. The authors introduce two functions, f(t) and h(t). f(t) gives a degree threshold: any n‑vertex graph whose average degree is at least f(t) contains a Kₜ‑model (a collection of connected subgraphs that can be contracted to the vertices of Kₜ) whose total number of vertices is bounded by h(t)·log n. The logarithmic dependence on n is shown to be optimal for fixed t, because there exist dense graphs in which any Kₜ‑model must use at least c·log n vertices.

The main quantitative result is that for every integer t≥2 and any ε>0, one can take f(t) ≤ 2^{t‑1}+ε. In other words, an average degree just above 2^{t‑1} already guarantees a Kₜ‑minor that can be realized with only O(log n) vertices. The authors also determine the exact thresholds for the smallest non‑trivial cases: f(3)=2+ε and f(4)=4+ε. These values improve upon the naïve bound 2^{t‑1} and match known lower bounds, showing that the results are tight for t≤4.

The proof strategy combines expansion properties of dense graphs with a careful construction of shallow breadth‑first search (BFS) trees. First, a graph with average degree ≥ f(t) is shown to contain a subgraph that is an (α,β)‑expander (every small vertex set has many external neighbours). Within such an expander, the authors grow t disjoint BFS trees of depth O(log n). Because of the expansion, each tree’s leaf set has a large neighbourhood that intersects the leaf sets of the other trees. By selecting a small collection of leaves from each tree and contracting each collection into a single “super‑vertex,” they obtain t super‑vertices that are pairwise adjacent after the contractions. The total number of original vertices involved in the construction is at most h(t)·log n, giving the desired bound.

A lower‑bound construction shows that the logarithmic factor cannot be removed: for fixed t, there exist families of graphs with average degree just above the threshold in which any Kₜ‑model must occupy at least c·log n vertices. This demonstrates that the authors’ bound is asymptotically best possible.

Beyond the unrestricted setting, the paper also treats graphs that can be embedded on a fixed surface (e.g., the torus or higher‑genus surfaces). By adapting the expansion arguments to respect the surface’s Euler characteristic, the authors prove analogous results: for each surface and each fixed t, a constant‑factor increase in the degree threshold still guarantees a Kₜ‑minor whose model uses only O(log n) vertices. This unifies the dense‑graph minor theory with topological graph theory.

Overall, the work provides a quantitative refinement of the “dense graphs contain large minors” paradigm. It shows that not only does a dense graph contain a large complete minor, but that the minor can be realized with a remarkably small, logarithmic‑size model. This insight has potential algorithmic implications for graph compression, kernelization, and the design of fixed‑parameter tractable algorithms that rely on finding small minors in large networks.