Abstract
AbstractIn this paper, we first investigate the total proper connection number of a graphGGaccording to some constraints ofG¯\overline{G}. Next, we investigate the total proper connection numbers of graphGGwith large clique numberω(G)=n−s\omega \left(G)=n-sfor1≤s≤31\le s\le 3. Finally, we determine the total properkk-connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.
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