On the sizes of bi-k-maximal graphs

  • Liqiong Xu
  • Yingzhi Tian
  • Hong-Jian LaiEmail author


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.
  1. (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 \(1\le r \le \lfloor \frac{n}{2k+2}\rfloor \).

  2. (ii)

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

  3. (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.



Edge connectivity Subgraph edge-connectivity Strength k-maximal graphs Bi-k-maximal graphs Uniformly dense graphs 

Mathematics Subject Classification

05C35 (05C40) 



The research of L. Xu is supported in part by National Natural Science Foundation of China (Nos. 11301217, 61572010), New Century Excellent Talents in Fujian Province University (No. JA14168) and Natural Science Foundation of Fujian Province, China (No. 2018J01419), Y. Tian is supported in part by National Natural Science Foundation of China (No. 11401510, 11861066) and Tianshan Youth Project (2018Q066), and H.-J. Lai is supported in part by National Natural Science Foundation of China (Nos. 11771093, 11771443).


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Authors and Affiliations

  1. 1.School of ScienceJimei UniversityXiamenChina
  2. 2.College of Mathematics and System SciencesXinjiang UniversityÜrümqiChina
  3. 3.Department of MathematicsWest Virginia UniversityMorgantownUSA

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