关于Murty-Simon猜想
On Murty-Simon Conjecture
摘要：
一个图是直径二边临界的，如果它的直径是2，并且删除任何边都会增加直径。Murty和Simon证明了$n$顶点上直径为2的边临界图的边数至多为$\lfloor \frac{n^{2}}{4} \rfloor$，其极图是完全二部图$K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ .在系列文献[8-10]中，Haynes等人提出的Murty-Simon猜想并不是最初的猜想，实际上，它只适用于直径为2的偶数阶临界图。Haynes等人证明了补图直径为3但顶点数为偶数的猜想。本文证明了补图直径为3的图的Murty-Simon猜想，其中补图的直径为3的图不仅有偶点，而且有奇点。
英文摘要：
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.
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