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이산수학 그룹
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Minimizing the number of edges in K(s,t)-saturated bipartite graphs

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Title
Minimizing the number of edges in K(s,t)-saturated bipartite graphs
Author(s)
DEBSOUMYA CHAKRABORTI; Chen, Da Qi; Hasabnis, Mihir
Subject
RESTRICTIONS
Publication Date
2021-01
Journal
SIAM JOURNAL ON DISCRETE MATHEMATICS, v.35, no.2, pp.1165 - 1181
Publisher
Society for Industrial and Applied Mathematics Publications
Abstract
© 2021 Society for Industrial and Applied Mathematics.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 Bollobas 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, Korandi, 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, Korandi, and Sudakov for general s and t, and for all sufficiently large n.
URI
https://pr.ibs.re.kr/handle/8788114/10041
DOI
10.1137/20M1368835
ISSN
0895-4801
Appears in Collections:
Pioneer Research Center for Mathematical and Computational Sciences(수리 및 계산과학 연구단) > Discrete Mathematics Group(이산 수학 그룹) > 1. Journal Papers (저널논문)
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