Find the edge ceiling

1 · Say what is proved

Beginner

What do Sanjana Das and Jacob Fox prove about diameter-2-critical graphs?

The bound, but only beyond a tower of 2's The bound for every number of vertices, from 3 up The bound, only for triangle-free graphs

2 · Work out the ceiling

The paper's key inequality: twice the number of edges is at most the number of pairs of vertices plus the number of edges in a maximal matching. A diameter-2-critical graph on 9 vertices has 36 pairs, and a matching there has at most 4 edges.

Intermediate

At most how many edges can this graph have?

18 20 22

3 · Explain the dead end

For an even number of vertices, GPT-5.6 Sol also proved that only the balanced complete bipartite graph reaches the ceiling, relying on a fact proved by Sumner about graphs in which every maximal matching is perfect, that is, covers every vertex.

Expert

Why could the authors not adapt this argument to an odd number of vertices?

The AI's bound was wrong for odd n The ceiling is out of reach for odd n With an odd count, no matching is perfect

4 · Put triangles to the test

Figure 1 shows a diameter-2-critical graph with 7 vertices and 9 edges that contains a triangle.

Expert

Can a diameter-2-critical graph containing a triangle ever reach the ceiling ⌊n²/4⌋?

No, never: a graph at the ceiling has no triangle Yes, when the number of vertices is odd Yes: Figure 1's graph does, 9 edges for 7 vertices

Ceiling found

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