The Number of Blocks of a Graph with Given Minimum Degree

Abstract

A block of a graph is a nonseparable maximal subgraph of the graph. We denote by $b\left(G\right)$ the number of block of a graph $G$ . We show that, for a connected graph $G$ of order $n$ with minimum degree $k\ge 1$ , $b\left(G\right)<\left(\left(2k-3\right)/\left(k^2-k-1\right)\right)n$ . The bound is asymptotically tight. In addition, for a connected cubic graph $G$ of order $n\ge 14$ , $b\left(G\right)\le \left(n/2\right)-2$ . The bound is tight.