A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and rainbow-connected there a between each pair its vertices. The minimum number colors needed to rainbow-connect G connection G, denoted by rc(G). simple way color spanning tree with distinct then re-use any these remaining G. This proves that rc(G)≤|V(G)|−1. We ask whether stronger tree-like structures colo...

The independence number α ( G ) $\alpha (G)$ and the dissociation diss $\text{diss}(G)$ of a graph $G$ are largest orders induced subgraphs maximum degree at most 0 1, respectively. We consider possible improvements obvious inequality 2 ≥ $2\alpha (G)\ge \text{diss}(G)$ . For connected cubic graphs distinct from K 4 ${K}_{4}$ , we show 5 3 $5\alpha 3\,\text{diss}(G)$ describe rich interesting s...