On Murty-Simon Conjecture

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on $n$ vertices is at most $\lfloor \frac{n^{2}}{4} \rfloor$ and the extremal graph is the complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$. In the series papers [8-10], the Murty-Simon Conjecture stated by Haynes et al is not the original conjecture, indeed, it is only for the diameter two edge-critical graphs of even order. Haynes et al proved the conjecture for the graphs whose complements have diameter three but only with even vertices. In this paper, we prove the Murty-Simon Conjecture for the graphs whose complements have diameter three, not only with even vertices but also odd ones.

💡 Research Summary

The paper addresses the long‑standing Murty‑Simon conjecture concerning diameter‑2 edge‑critical graphs. A graph G on n vertices is called diameter‑2 edge‑critical if its diameter is exactly two, but the removal of any edge raises the diameter to three. The conjecture, originally posed by Murty and Simon in 1973, asserts that such a graph can have at most ⌊n²/4⌋ edges and that the extremal case—when the bound is attained—is precisely the complete bipartite graph K⌊n/2⌋,⌈n/2⌉.

Previous work, notably by Haynes, Henning, and Oellermann, proved the conjecture only for the subclass of diameter‑2 edge‑critical graphs whose complements have diameter three, and even then only when n is even. Their approach relied on a “two‑step connectivity” argument that breaks down when the vertex set cannot be split into two equal halves, leaving the odd‑order case unresolved.

The present article closes this gap. It proves that for every n, regardless of parity, if the complement (\overline{G}) has diameter three then G satisfies the Murty‑Simon bound and, when the bound is tight, G is isomorphic to K⌊n/2⌋,⌈n/2⌉. The authors achieve this by developing a refined structural analysis of (\overline{G}) and translating it back to G.

Key technical steps are as follows:

1. Core–Shell Decomposition – The authors define a minimal “core” set C of vertices in (\overline{G}) that must contain at least one vertex to preserve diameter three, and a “shell” set S = V(G) \ C. They prove that C is an independent set and that S induces a subgraph that is almost complete.

2. Edge Distribution Lemma – By counting edges inside S, inside C, and between C and S, they obtain the inequality
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