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Minimizing the number of edges in $K_{s,t}$-saturated bipartite graphs
- Publication Year :
- 2020
- Publisher :
- arXiv, 2020.
-
Abstract
- This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollob\'as independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered' complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural `unordered' analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Kor\'andi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in $K_{s,t}$-saturated $n$ by $n$ bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for $s=t-1$ with the classification of the extremal graphs. We also improve the estimates of Gan, Kor\'andi, and Sudakov for general $s$ and $t$, and for all sufficiently large $n$.<br />Comment: Reflected minor suggestions from reviewers
- Subjects :
- Discrete mathematics
Mathematics::Combinatorics
General Mathematics
Minimization problem
0102 computer and information sciences
Edge (geometry)
01 natural sciences
Complete bipartite graph
Combinatorics
Computer Science::Discrete Mathematics
010201 computation theory & mathematics
TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY
Bipartite graph
FOS: Mathematics
Mathematics - Combinatorics
Combinatorics (math.CO)
MathematicsofComputing_DISCRETEMATHEMATICS
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....1e13440192987cb8661470bf764f0846
- Full Text :
- https://doi.org/10.48550/arxiv.2009.07651