Improving the Bounds On Murty_Simon Conjecture

TL;DR

This paper improves the upper bound on the degree condition for the Murty-Simon conjecture, showing it holds for all sufficiently large graphs when the maximum degree is at least 0.676 times the number of vertices.

Contribution

The authors tighten the degree bound for the Murty-Simon conjecture from 0.6789n to 0.676n, extending its validity to all graphs beyond a certain size.

Findings

The conjecture holds for graphs with maximum degree ≥ 0.676n.

Previous bounds were 0.6789n for large n and 0.7n generally.

The new bound applies to all sufficiently large graphs.

Abstract

A graph is said to be diameter-$k$-critical if its diameter is $k$ and removal of any of its edges increases its diameter. A beautiful conjecture by Murty and Simon, says that every diameter-2-critical graph of order $n$ has at most $\lfloor n^2/4\rfloor$ edges and equality holds only for $K_{\lceil n/2 \rceil,\lfloor n/2 \rfloor }$. Haynes et al. proved that the conjecture is true for $\Delta\geq 0.7n$. They also proved that for $n>2000$, if $\Delta \geq 0.6789n$ then the conjecture is true. We will improve this bound by showing that the conjecture is true for every $n$ if $\Delta\geq\ 0.676n$.