No, that is incorrect. Diamond does not split, and a pure Diamond Dragon can be bred with any 4 colors, not just the ones you listed. Purple does not ever split into red and blue during breeding (as claimed, above). It only contributes purple to the breeding pool.
Only minor types (pink, black, tropic, dino, metal, etc.) can split. Since the pure purple dragons (Magic or Peacock) only ever contribute purple to the breeding pool, purple it is not considered a minor type. Even when the parent is a purple hybrid (like Mistmoth or Genie), the purple never splits.
An example: Genie X Forest can produce purple/yellow dragons, purple/greens, green/yellows, purple/green/yellows, and the pure Fairytale Dragon (from yellow in one parent, green in the other). No blues or reds to be found since purple never splits.
The pure diamond dragons (the Diamond and Infinity Dragon) do have their own, unique behaviors as a parents, but they never split into other colors. The diamond-type in a diamond hybrid (like in the Crusader or Mercury Dragon) also does not ever split. The diamond-type in diamond hybrids behaves like a primary color.
An example: Mercury X Air only has 13 outcomes: all 9 of the blue/yellow hybrid dragons, the 2 diamond/blues, the 1 diamond/yellow, and the pure Rainbow dragon (from blue in one parent, yellow in the other). The diamond-type acts just like a primary color.
Diamond X Infinity is a special combo because it can produce any pure-type dragon (Fire Dragon, Water Dragon, Fairytale Dragon, Metal Dragon, Cosmic Dragon, Air Dragon, Stone Dragon, Peacock Dragon, etc.) But, that unique behavior is is not a split. Breeding Diamond/Infinity with hybrids has more complex results- it all depends on the hybrid's type(s). It's too much, there are too many variables, to try and explain it all here.
I highly recommend checking out and playing around with one of the breeding calculators to better understand how breeding works. The rules of parents' behavior are also listed, in detail, on other sites.
As for the original question: yes, it is possible for one parent to not contribute anything to the offspring.
For example, when breeding Atlantis (green/blue) X Air (yellow), you can get any of the green/blue dragons as your outcome.
(But, you can't get a Gemstone Dragon from this combo because the blue and the green are both on the same side. If you bred a Forest (green) X Mist (blue/yellow), for example, you could get a Gemstone Dragon because the blue and the green are now on opposite sides.)