Graphs with strong 3-rainbow index equals 2

Abstract

AbstractLetGbe an edge-colored connected graph of ordern≥ 3, where adjacent edges may be colored the same. Letkbe an integer with 2 ≤k≤nandS⊆V(G) with |S| =k. The Steiner distanced(S) ofSis the minimum size of a tree inGconnectingS. The strongk-rainbow indexsrxk(G) ofGis the minimum number of colors required to color the edges ofGso that every setSinGis connected by a tree of sized(S) whose edges have distinct colors. We focus onk= 3. In this paper, we first characterize the graphsGwithsrx3(G) = 2. According to the definition, it is clearly that ‖G‖ is the trivial upper bound forsrx3(G). Several previous researchers have shown that there exist some connected graphsGsuch thatsrx3(G) = ‖G‖. Hence, in this paper, we provide another graphGsuch thatsrx3(G) = ‖G‖.