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

Abstract

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 rc1,2(G), is referred to as the (1,2)-rainbow connection number. If G is a nontrivial connected graph of size m, then rc1,2(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≤rc1,2(G)≤m−1.