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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This work was supported by the NSFC (No. 11701157 and No. 61907014), and the Foundation of Henan Normal University (No. 2020PL05 and No. 2019QK06)
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Ma, Y., Nie, K. (Strong) Total proper connection of some digraphs. J Comb Optim 42, 24–39 (2021). https://doi.org/10.1007/s10878-021-00738-w
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DOI: https://doi.org/10.1007/s10878-021-00738-w