Let $G$ a graph such that $|V(G)| = p, \Delta(G) = p-1$ and $\delta_G(x) = \Delta(G)$ for some $x \in V(G)$. Show that if $e$ is a bridge of $G$ then $x$ is an endpoint of $e$.
I have so many doubts about if this is provable, because I think that a bridge could be independent of a cut vertex. But anyway, do you have any hints for this?
Might as well convert to an answer.
If a graph has $n$ vertices and contains a vertex $x$ of degree $n - 1$, then every other vertex is adjacent to $x$. That means, any two vertices are connected by a path of at most length $2$, via the vertex $x$.
If you remove an edge that is not adjacent to $x$, then you still have a graph of $n$ vertices with one vertex of degree $n - 1$, hence it will still be connected. Therefore, removing an edge not adjacent to $x$ cannot disconnect the graph, and hence every bridge must be adjacent to $x$.