A square matrix of order
![$n$](img15.png)
is called a Toeplitz matrix if it has
constant values along all diagonals parallel to the main diagonal. A
directed Toeplitz graph
![$T_n\langle
s_1,\dots,s_k;t_1,\dots,t_l\rangle$](img16.png)
with vertices
![$1, 2, \dots, n$](img17.png)
,
where the edge
![$(i,\,j)$](img18.png)
occurs if and only if
![$j-i=s_p$](img19.png)
or
![$i-j=t_q$](img20.png)
for some
![$1\leq p\leq k$](img21.png)
and
![$1\leq q\leq l$](img22.png)
, is a digraph
whose adjacency matrix is a Toeplitz matrix. In this paper, we study
hamiltonicity in directed Toeplitz graphs
![$T_n\langle 1, 3; 1,
t\rangle$](img14.png)
. We obtain new results and improve existing results on
![$T_n\langle 1, 3; 1,
t\rangle$](img14.png)
.