4- regular triangle free connected graph I know that there exists a $4$-regular connected graph with even number of vertices that does not have a perfect matching ( an example of this graph is stated here)
I want to know if there is a $4$-regular triangle free connected graph with even number of vertices that does not have a perfect matching? or if not, how can I prove it by Tutte's theorem?
 A: Yes, the example you linked can be modified to be triangle free and still be $4$-regular, have an even number of vertices, and have no perfect matching. 
Just replace each component $Ci$ with a $4$-regular graph with large girth and odd number of vertices.
The total number of vertices will be $2 + 4\cdot \#V(C1)$ which is even,
and the graph has no perfect matching by the same argument.
Edit: To be more specific, you could take each of the components $C1,\ldots,C4$ to be this graph:

This graph has 11 vertices and is triangle free, since the shortest cycles have length 4.
In general, the girth of a graph means the minimal length of a cycle.
There exist $4$-regular graphs with arbitrarily large girth $g$.
The ones with fewest possible vertices are known as $(4,g)$ cage graphs.
We can generalize the example above by  choosing each $Ci$ to be a $(4,g)$ cage graph with an odd number of vertices, e.g. 
the Robertson graph,
but with one edge ''broken up'' to connect to the vertices in the middle component $S$.
