Stefan Glock, Felix Joos, Jaehoon Kim, Marcus Kühn, Lyuben Lichev
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We say that a matching <span></span><math>\n <semantics>\n <mrow>\n <mi>M</mi>\n <mo>⊆</mo>\n <mi>E</mi>\n <mo>(</mo>\n <mi>H</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {M}\\subseteq E(\\mathcal {H})$</annotation>\n </semantics></math> is conflict-free if <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math> does not contain an element of <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$\\mathcal {C}$</annotation>\n </semantics></math> as a subset. Under natural assumptions on <span></span><math>\n <semantics>\n <mi>C</mi>\n <annotation>$\\mathcal {C}$</annotation>\n </semantics></math>, we prove that <span></span><math>\n <semantics>\n <mi>H</mi>\n <annotation>$\\mathcal {H}$</annotation>\n </semantics></math> has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called ‘high-girth’ Steiner systems. 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We extend this result by obtaining a <i>conflict-free</i> matching, where conflicts are encoded via a collection <span></span><math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>$\\\\mathcal {C}$</annotation>\\n </semantics></math> of subsets <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>C</mi>\\n <mo>⊆</mo>\\n <mi>E</mi>\\n <mo>(</mo>\\n <mi>H</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$C\\\\subseteq E(\\\\mathcal {H})$</annotation>\\n </semantics></math>. We say that a matching <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>M</mi>\\n <mo>⊆</mo>\\n <mi>E</mi>\\n <mo>(</mo>\\n <mi>H</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathcal {M}\\\\subseteq E(\\\\mathcal {H})$</annotation>\\n </semantics></math> is conflict-free if <span></span><math>\\n <semantics>\\n <mi>M</mi>\\n <annotation>$\\\\mathcal {M}$</annotation>\\n </semantics></math> does not contain an element of <span></span><math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>$\\\\mathcal {C}$</annotation>\\n </semantics></math> as a subset. Under natural assumptions on <span></span><math>\\n <semantics>\\n <mi>C</mi>\\n <annotation>$\\\\mathcal {C}$</annotation>\\n </semantics></math>, we prove that <span></span><math>\\n <semantics>\\n <mi>H</mi>\\n <annotation>$\\\\mathcal {H}$</annotation>\\n </semantics></math> has a conflict-free, almost-perfect matching. 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引用次数: 0
摘要
皮彭格(Pippenger)、弗兰克尔(Frankl)和罗德尔(Rödl)的一个著名定理指出,每一个几乎不规则的、具有较小最大度数的均匀超图 H $mathcal {H}$ 都有一个几乎完美的匹配。我们通过获得无冲突匹配来扩展这一结果,其中冲突是通过子集 C ⊆ E ( H ) $C\subseteq E(\mathcal {H})$ 的集合 C $\mathcal {C}$ 来编码的。如果 M $\mathcal {M}$ 不包含作为子集的 C $\mathcal {C}$ 的元素,我们就说匹配 M ⊆ E ( H ) $\mathcal {M}\subseteq E(\mathcal {H})$ 是无冲突的。在 C $\mathcal {C}$ 的自然假设下,我们证明 H $\mathcal {H}$ 有一个无冲突的、几乎完美的匹配。这一点有很多应用,其中之一是为所谓的 "高出生 "斯坦纳系统提供了新的渐近结果。我们的主要工具是一种随机贪婪算法,我们称之为 "无冲突匹配过程"。
A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost-regular, uniform hypergraph with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a conflict-free matching, where conflicts are encoded via a collection of subsets . We say that a matching is conflict-free if does not contain an element of as a subset. Under natural assumptions on , we prove that has a conflict-free, almost-perfect matching. This has many applications, one of which yields new asymptotic results for so-called ‘high-girth’ Steiner systems. Our main tool is a random greedy algorithm which we call the ‘conflict-free matching process’.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.