Some results on the total proper<i>k</i>-connection number
Some results on the total proper<i>k</i>-connection number
Abstract In this paper, we first investigate the total proper connection number of a graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>G</m:mi></m:math> G according to some constraints of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mover accent="true"><m:mrow><m:mi>G</m:mi></m:mrow><m:mrow><m:mo stretchy="true">¯</m:mo></m:mrow></m:mover></m:math> \overline{G} . Next, we investigate the total proper connection numbers of graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>G</m:mi></m:math> G with large clique number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ω</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>G</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>n</m:mi><m:mo>−</m:mo><m:mi>s</m:mi></m:math> \omega \left(G)=n-s …