Completing partial packings of bipartite graphs

Füredi, Zoltán; Riet, Ago-Erik; Tyomkyn, Mykhaylo

doi:10.1016/j.jcta.2011.06.009

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Proof Of Theorem 141

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2021

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“…Also, note that m(G) = m(G ). Hence, as t − γ(n + t) < i ≤ u, we have from ( 7) that (8) m(G) ≥ min{f (t − γ(n + t)), f (u)}. Regardless of the parity of n + t, using ( 8)- (12) we conclude that (6) holds.…”

Section: Proof Of Theorem 14mentioning

confidence: 81%

On deficiency problems for graphs

Freschi

Hyde

Treglown

2021

Combinator. Probab. Comp.

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Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$ , the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer t such that the join $G*K_t$ has property $\mathcal P$ . In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$ -factor (for any fixed $r\geq 3$ ). In this paper, we resolve their problem fully. We also give an analogous result that forces $G*K_t$ to contain any fixed bipartite $(n+t)$ -vertex graph of bounded degree and small bandwidth.

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Section: Proof Of Theorem 14mentioning

confidence: 81%