In a graph, if every group of $4$ vertices has a vertex that is connected to all the other $3$ vertices can we say there exists a vertex connected to all the other vertices in the graph? Note that the number of vertices of the graph $\ge4.$
My approach: I have tried drawing small graphs and applying the given rules on those. But for every instance, there exists a vertex connected to all other $n-1$ vertices. I am not sure how to prove this.
If every two vertices are connected we are done.
So suppose there is a pair $A,B$ not connected.
Now take new pair $X,Y$. Then (we observe group $\{A,B,X,Y\}$, one of them is connected with all of them in group, cleary that can not be $A$ or $B$ since they are not connected) $X$ are $Y$ are connected and one of them is connected to $A$, $B$. So all vertices in $G\setminus \{A,B\}$ are connected and some of them are also connected to $A$ and $B$.
So if $n\geq 4$ there is a vertex connected to all other vertices.