On the sizes of bi-k-maximal graphs

Let k , n , s , t > 0 be integers and n = s + t ≥ 2 k + 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 } ≥ k on n vertices, then each of the following holds. Let m be an integer. Then there exists a bi- k -maximal graph G with m = | E ( G ) | if and only if m = n k - r k 2 + ( r - 1 ) k for some integer r with 1 ≤ r ≤ ⌊ n 2 k + 2 ⌋ . Every bi- k -maximal graph G on n vertices satisfies | E ( G ) | ≤ ( n - k ) k , and this upper bound is tight. Every bi- k -maximal graph G on n vertices satisfies | E ( G ) | ≥ k ( n - 1 ) - ( k 2 - k ) ⌊ n 2 k + 2 ⌋ , and this lower bound is tight. Moreover, the bi- k -maximal graphs reaching the optimal bounds are characterized.