Some results on the total proper k-connection number

Abstract

In this paper, we first investigate the total proper connection number of a graph GG according to some constraints of G¯\overline{G}. Next, we investigate the total proper connection numbers of graph GG with large clique number ω(G)=n−s\omega \left(G)=n-s for 1≤s≤31\le s\le 3. Finally, we determine the total proper kk-connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.

Keywords

• total coloring
• total proper path
• total proper k-connected
• total proper k-connection number
• complement graph
• clique number
• 05c15
• 05c35
• 05c40