Affiliation:
1. College of Mathematics and Information Science, Henan Normal University , Xinxiang 453007 , P. R. China
Abstract
Abstract
In this paper, we first investigate the total proper connection number of a graph
G
G
according to some constraints of
G
¯
\overline{G}
. Next, we investigate the total proper connection numbers of graph
G
G
with large clique number
ω
(
G
)
=
n
−
s
\omega \left(G)=n-s
for
1
≤
s
≤
3
1\le s\le 3
. Finally, we determine the total proper
k
k
-connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.