Edge Growth in Graph Cubes

We show that for every connected graph $G$ of diameter $\ge 3$, the graph $G^3$ has average degree $\ge 7/4 \delta(G)$. We also provide an example showing that this bound is best possible. This resolves a question of Hegarty \cite{PH}.

💡 Research Summary

The paper investigates how the number of edges grows when a graph is raised to its third power. For a simple, connected graph G, the k‑th power Gᵏ is obtained by adding an edge between any two vertices whose distance in G is at most k. The authors focus on the case k = 3 and on graphs whose diameter is at least 3, because only then does G³ differ non‑trivially from G.

Earlier work by Hegarty showed that for every connected regular graph of diameter at least 3 there exists a positive constant c such that e(G³) ≥ (1 + c) e(G). Pokrovskiy later improved the known lower bound to c = 1⁄6. However, the exact optimal constant remained open. The present paper resolves this question completely.

Main Result (Theorem 1.3).
If G is any connected graph with diameter ≥ 3, then
 e(G³) ≥ (7⁄8) δ(G) · v(G),
where δ(G) is the minimum degree and v(G) the number of vertices. Since the average degree of a graph equals 2e/v, this inequality is equivalent to the statement that the average degree of G³ is at least (7⁄4) δ(G). In particular, for regular graphs the bound reads e(G³) ≥ (1 + 3⁄4) e(G), i.e. the constant c can be taken as 3⁄4, which is the best possible.

Proof Overview.
The authors introduce the notion of a doubling vertex: a vertex v for which deg_{G³}(v) ≥ 2 δ(G). Let Z be the set of all doubling vertices. Removing Z from G leaves a collection of connected components X₁,…,X_m. Several auxiliary claims are proved:

1. Any internal vertex of a geodesic (shortest) path of length 3 is a doubling vertex.
2. If two vertices belong to the same component X_i, then their distance‑2 neighborhoods coincide.
3. If two distinct components X_i and X_j have disjoint neighborhoods (N(X_i)∩N(X_j)=∅), then any vertex from X_i and any vertex from X_j also have identical distance‑2 neighborhoods.

Using (2) and (3) the authors define an equivalence relation ∼ on the components, grouping them into equivalence classes Y₁,…,Y_ℓ. For each class Y_i, the set N(Y_i) induces a clique in G². Because the overall diameter is at least 3, G² cannot be a complete graph, so there exists a vertex u at distance 2 from Y_i but not adjacent to any vertex of Y_i. This vertex together with Y_i yields the inequality deg_{G³}(v) ≥ δ + |Y_i| for every v ∈ Y_i.

Next, they bound the size of Z. Each component Y_i contributes at least δ − |Y_i| vertices of Z that are adjacent to Y_i, and these contributions are disjoint across different Y_i. Hence |Z| ≥ δ·ℓ − y, where y = ∑|Y_i|.

Finally, summing the degree lower bounds over all vertices gives
∑{v∈V} deg{G³}(v) − (7⁄4)δ·v(G) ≥ (1⁄4)δ·|Z| − (3⁄4)δ·y + ∑|Y_i|² ≥ 0,
where the last inequality follows from the previous estimate on |Z| and the elementary inequality a² + b² ≥ 2ab. This establishes the desired average‑degree bound.

Sharpness Construction.
To show that the constant 7⁄8 cannot be improved, the authors construct a family {G_k} as follows. Take five subgraphs H₁,…,H₅: H₁ and H₅ are copies of K_{2k+1}; H₂ and H₄ are copies of K_{2k} with a perfect matching removed; H₃ is a single isolated vertex. Connect each H_i to H_{i+1} by adding all possible edges between them (for i = 1,…,4). The resulting graph G_k is (4k)-regular, has 8k + 3 vertices, and its third power G_k³ has exactly (7⁄8)·δ·v edges, confirming that the bound is tight.

Additional Remarks.
The paper also proposes a conjecture for directed graphs (Conjecture 1.4): if D is an orientation of a simple graph in which every vertex has indegree = outdegree = d, then e(D²) ≥ 2 e(D). This conjecture, if true, would resolve a special case of the well‑known Caccetta‑Häggkvist conjecture.

Conclusion.
By removing the regularity assumption and proving a sharp lower bound on the edge count of the third power of any connected graph with diameter at least three, the authors settle a question raised by Hegarty. The proof combines careful neighborhood analysis, an equivalence‑class decomposition, and a tight extremal construction, thereby advancing our understanding of graph power growth and its connections to additive combinatorics and group theory.