Abstract
The total proper connection number of a given digraph D, represented by $$\overrightarrow{tpc}(D)$$ , denotes the smallest number of colors needed for making D total proper connected. The strong total proper connection number of D, represented by $$\overrightarrow{stpc}(D)$$ , shows the smallest number of colors required for making D strong total proper connected. In the present work, we represent some preliminary findings on $$\overrightarrow{tpc}(D)$$ and $$\overrightarrow{stpc}(D)$$ . Moreover, findings on the (strong) total proper connection numbers of biorientations of graphs, circle digraphs, circulant digraphs and cacti digraphs are provided.
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