The bipartiteness and maximum-degree conjecture for mathcal{C}_3-free BonnetMyers sharp graphs

Let $G$ be a graph that is $\mathcal{C}_3$-free and BonnetMyers sharp, with diameter $L$. For each vertex $u$, let $d_u$ denote its degree.

Bipartiteness and maximum-degree conjecture.

1. $G$ is bipartite.
2. For every edge $uv$,

$$\max\{d_u,d_v\}=L.$$

The conjecture extends the authors' classification of $\mathcal{C}_3$-free BonnetMyers sharp graphs with diameters $L=2,3,4,5$. The stated pattern is observed in those classified examples, but its validity for all such graphs remains open.