A path in an edge-colored graph is called a monochromatic path if all edges of the path have a same color. We call k paths P1,⋯,Pk rainbow monochromatic paths if every Pi
Nearby in the stack
is monochromatic and for any two
i=j
,
Pi
and
Pj
have different colors. An edge-coloring of a graph
G
is said to be a rainbow monochromatic
k
-edge-connection coloring (or
RMCk
-coloring for short) if every two distinct vertices of
G
are connected by at least
k
rainbow monochromatic paths. We use
rmck(G)
to denote the maximum number of colors that ensures
G
has an
RMCk
-coloring, and this number is called the rainbow monochromatic
k
-edge-connection number. We prove the existence of
RMCk
-colorings of graphs, and then give some bounds of
rmck(G)
and present some graphs whose
rmck(G)
reaches the lower bound. We also obtain the threshold function for
rmck(G(n,p))≥f(n)
, where
⌊2n⌋>k≥1
.
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