On the sizes of bi-k-maximal graphs

Abstract

Let $k,n, s, t > 0$ be integers and $n = s+t \ge 2k+2$. A simple bipartite graph G spanning $K_{s,t}$ is bi-k-maximal, if every subgraph of G has edge-connectivity at most k but any edge addition that does not break its bipartiteness creates a subgraph with connectivity at least $k+1$. We investigate the optimal size bounds of the bi-k-maximal simple graphs, and prove that if G is a bi-k-maximal graph with $\min \{s, t \} \ge k$ on n vertices, then each of the following holds.

• (i) Let m be an integer. Then there exists a bi-k-maximal graph G with $m = |E(G)|$ if and only if $m = nk - rk^2 + (r-1)k$ for some integer r with $1\le r \le \lfloor \frac{n}{2k+2}\rfloor$.

• (ii) Every bi-k-maximal graph G on n vertices satisfies $|E(G)| \le (n-k)k$, and this upper bound is tight.

• (iii) Every bi-k-maximal graph G on n vertices satisfies $|E(G)| \ge k(n-1) - (k^2-k)\lfloor \frac{n}{2k+2}\rfloor$, and this lower bound is tight. Moreover, the bi-k-maximal graphs reaching the optimal bounds are characterized.