The structure of minimal weight rainbow domination functions of cubic graphs are studied. Based on general observations for cubic graphs, generalized Petersen graphs $P(ck,k)$ are characterized whose 4- and 5-rainbow domination numbers equal the general lower bounds. As t-rainbow domination of cubic graphs for t ≥ 6 is trivial, characterizations of such generalized Petersen graphs $P(ck,k)$ are known for all t-rainbow domination numbers.In addition, new upper bounds for 4- and 5-rainbow domination numbers that are valid for all $P(ck,k)$ are provided.
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Janez Žerovnik (2024) studied this question.
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