For a graph G , let γ r 2 ( G ) and γ R ( G ) denote the 2-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (2013) proved γ r 2 ( G ) + γ R ( G ) ? 6 4 n ( G ) for a connected graph G of order n ( G ) at least 3. Furthermore, they conjectured γ r 2 ( G ) + γ R ( G ) ? 4 3 n ( G ) for a connected graph G of minimum degree at least 2 that is distinct from C 5 . We characterize all extremal graphs for their inequality and prove their conjecture.
['José D. Alvarado', 'Simone Dantas', 'Dieter Rautenbach']
Averaging 2-rainbow domination and Roman domination