A note on rainbow saturation number of paths

Abstract For a fixed graph F and an integer t, the rainbow saturation number of F, denoted by s a t t ( n , R ( F ) ) , is defined as the minimum number of edges in a t-edge-colored graph on n vertices which does not contain a rainbow copy of F, i.e., a copy of F all of whose edges receive a different color, but the addition of any missing edge in any color from [t] creates such a rainbow copy. Barrus, Ferrara, Vardenbussche and Wenger prove that s a t t ( n , R ( P l ) ) ≥ n − 1 for l ≥ 4 and s a t t ( n , R ( P l ) ) ≤ ⌈ n l − 1 ⌉ · ( l − 1 2 ) for t ≥ ( l − 1 2 ) , where Pl is a path with l edges. In this short note, we improve the upper bounds and show that s a t t ( n , R ( P l ) ) ≤ ⌈ n l ⌉ · ( ( l − 2 2 ) + 4 ) for l ≥ 5 and t ≥ 2 l − 5 .