Graphs with large (1,2)-rainbow connection numbers.

Let G be an edge-colored graph. If every subpath of length at most l + 1 within a path P in a graph G consists of uniquely colored edges, then P is called an l -rainbow path. A connected graph G is deemed (1 , 2) -rainbow connected if there exists at least one 2-rainbow path connecting two distinct vertices within G. The minimum number of colors needed to attain (1 , 2) -rainbow connectedness in a connected graph G , represented as r c 1 , 2 (G) , is referred to as the (1 , 2) -rainbow connection number. If G is a nontrivial connected graph of size m , then r c 1 , 2 (G) = m if and only if G is the star or double star of size m. Our main goal is to identify all connected graphs G of size m that satisfy the condition m − 3 ≤ r c 1 , 2 (G) ≤ m − 1. [ABSTRACT FROM AUTHOR]