New Bounds on the Triple Roman Domination Number of Graphs

In this paper, we derive sharp upper and lower bounds on the sum gamma([3R])(G)+gamma([3R])(G over bar) and product gamma([3R])(G)gamma([3R])(G over bar), where G over bar is the complement of graph G. We also show that for each tree T of order n & GE;2, gamma([3R])(T)& LE;3n+sT/2 and gamma([3R])(T)& GE; left ceiling 4(n(T)+2-l(T))/3 right ceiling , where s(T) and l(T) are the number of support vertices and leaves of T.