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# On the sizes of bi-k-maximal graphs

• Liqiong Xu
• Yingzhi Tian
• Hong-Jian Lai
Article

## 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.
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.

## Keywords

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

05C35 (05C40)

## Notes

### Acknowledgements

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).

## References

1. Anderson J (2017) A study of arc strong connectivity of digraphs, PhD Dissertation, West Virginia UniversityGoogle Scholar
2. Anderson J, Lai H-J, Lin X, Xu M (2017) On $$k$$-maximal strength digraphs. J Graph Theory 84:17–25
3. Anderson J, Lai H-J, Li X, Lin X, Xu M (2018) Minimax properties of some density measures in graphs and digraphs. Int J Comput Math Comput Syst Theor 3(1):1–12
4. Bondy JA, Murty USR (2008) Graph theory. Springer, New York
5. Chen Y-C, Tan JJM, Hsu L-H, Kao S-S (2003) Super-connectivity and super-edge-connectivity for some internation networks. Appl Math Comput 140:245–254
6. Esfahanian AH, Hakimi SL (1988) On computing a conditional edgeconnectivity of a graph. Inf Process Lett 27:195–199
7. Lai H-J (1990) The size of strength-maximal graphs. J Graph Theory 14:187–197
8. Li P, Lai H-J, Xu M (2019) Disjoint spanning arborescences in $$k$$-arc-strong digraphs. ARS Comb 143:147–163
9. Lin X, Fan S, Lai H-J, Xu M (2016) On the lower bound of $$k$$-maximal digraphs. Discrete Math 339:2500–2510
10. Mader W (1971) Minimale $$n$$-fach kantenzusammenhngende graphen. Math Ann 191:21–28
11. Matula DW (1968) A min–max theorem for graphs with application to graph coloring. SIAM Rev 10:481–482Google Scholar
12. Matula DW (1969) The cohesive strength of graphs. In: Chartrand G, Kapoor SF (eds) The many facets of graph theory. Lecture notes in mathematics, vol 110. Springer, Berlin, pp 215–221
13. Matula D (1972) $$K$$-components, clusters, and slicings in graphs. SIAM J Appl Math 22:459–480
14. Matula DW (1976) Subgraph connectivity number of a graph. In: Alavi Y, Ricks DR (eds) Theory and applications of graphs. Lecture notes in mathematics, vol 642. Springer, Berlin, pp 371–393
15. Xu JM. Super or restricted connectivity of graphs, a survey (to appear) Google Scholar
16. Xu M (2018) A study on graph coloring and digraph connectivity, PhD Dissertation, West Virginia UniversityGoogle Scholar

## Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

## Authors and Affiliations

• Liqiong Xu
• 1
• Yingzhi Tian
• 2
• Hong-Jian Lai
• 3
1. 1.School of ScienceJimei UniversityXiamenChina
2. 2.College of Mathematics and System SciencesXinjiang UniversityÜrümqiChina
3. 3.Department of MathematicsWest Virginia UniversityMorgantownUSA