|Technically speaking there is no such thing as an Ethernet cable.|
Ethernet is a network protocol used on Local Area Networks, basically Software.
Cat5 and Cat6 are cable specifications for use on a netwrok using the Ethernet Protocol.
What do you mean by a 6 port splitter. Do you mean a six port switch? If so how are you connecting them? You only need to connect one cable from the router to the switch via its up-link port if it has one and any other devices to the other ports. Your use of the word 'them' which is plural has me a bit confused.
Cat5/6 cables comes in two varieties, patch cable and crossover. The difference is in the way the plugs are wired. Normally you would connect the router to the switch via a crossover cable but as most switches these days have auto sensing ports it is not always necessary.
Knowing the model number of this device you call a splitter would be an enormous help.