On the sizes of bi-k-maximal graphs

Liqiong Xu (), Yingzhi Tian () and Hong-Jian Lai ()
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Liqiong Xu: Jimei University
Yingzhi Tian: Xinjiang University
Hong-Jian Lai: West Virginia University

Journal of Combinatorial Optimization, 2020, vol. 39, issue 3, No 12, 859-873

Abstract: Abstract Let $$k,n, s, t > 0$$k,n,s,t>0 be integers and $$n = s+t \ge 2k+2$$n=s+t≥2k+2. A simple bipartite graph G spanning $$K_{s,t}$$Ks,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 $$k+1$$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 $$\min \{s, t \} \ge k$$min{s,t}≥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 $$m = |E(G)|$$m=|E(G)| if and only if $$m = nk - rk^2 + (r-1)k$$m=nk-rk2+(r-1)k for some integer r with $$1\le r \le \lfloor \frac{n}{2k+2}\rfloor $$1≤r≤⌊n2k+2⌋.(ii)Every bi-k-maximal graph G on n vertices satisfies $$|E(G)| \le (n-k)k$$|E(G)|≤(n-k)k, and this upper bound is tight.(iii)Every bi-k-maximal graph G on n vertices satisfies $$|E(G)| \ge k(n-1) - (k^2-k)\lfloor \frac{n}{2k+2}\rfloor $$|E(G)|≥k(n-1)-(k2-k)⌊n2k+2⌋, 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) (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-020-00522-2

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