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arXiv 1902.00682math.CO
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向量团分解

Vector clique decompositions

Raphael Yuster

备注：
30 pages
发表信息：
Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, 1221-1238
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摘要：

设$F_k$是$k$顶点上的图的集合。对于图$G$，$k$ -分解是$G$的导出子图的集合，每个导出子图同构于$F_k$的一个元素，使得$G$的每对顶点恰好在集合的一个元素中。威尔逊的一个基本结果是，对于所有足够大的$n=|V(G)|$，$G$有$k$ -分解当且仅当$G$是$k$ -可分的。令${\bf v} \in {\mathbb R}^{|F_k|}$由$F_k$索引。对于$G$的$k$ -分解$L$，设$ν_{\bf v}(L) = \sum_{F \in F_k} {\bf v}_F d_{L,F}$，其中$d_{L,F}$是与$F$同构的$L$的元素的分数。设$ν_{\bf v}(G) = \max_{L} ν_{\bf v}(L)$和$ν_{\bf v}(n)=\min\{ν_{\bf v}(G):|V(G)|=n\}$。证明序列$ν_{\bf v}(n)$具有极限并不困难，因此设$ν_{\bf v} = \lim_{n \rightarrow \infty} ν_{\bf v}(n)$。用它们的分数弛豫代替$k$ -分解，得到（多项式时间可计算的）分数模拟$ν_{\bf v}^*(G)$和相应的分数值$ν^*_{\bf v}(n)$和$ν^*_{\bf v}$。我们的第一个主要结果是，对于每个${\bf v} \in {\mathbb R}^{|F_k|}$ $$ ν_{\bf v} = ν^*_{\bf v}\;. $$此外，有一个多项式时间的算法，产生一个$k$分解$L$-可分解图，使$ν_{\bf v}(L) \ge ν_{\bf v} - o_n(1)$。当$F_k$是$k$顶点上的所有竞赛图的族时，以及当$F_k$是$K_k$的所有边着色的族时，类似的结果也成立。我们使用这些结果，以获得新的和改进的几个分解结果的界限。例如，我们证明了每个$n$ -顶点竞赛是$3$ -整除有一个三角形分解，其中有向三角形的数目小于$0.0222n^2(1+o(1))$，每个$5$ -可分解的$n$ -顶点图有一个$5$ -分解，其中长度为$5$的循环的分数是$o_n(1)$。

英文摘要：

Let $F_k$ be the set of graphs on $k$ vertices. For a graph $G$, a $k$-decomposition is a set of induced subgraphs of $G$, each isomorphic to an element of $F_k$, such that each pair of vertices of $G$ is in exactly one element of the set. A fundamental result of Wilson is that for all $n=|V(G)|$ sufficiently large, $G$ has a $k$-decomposition if and only if $G$ is $k$-divisible. Let ${\bf v} \in {\mathbb R}^{|F_k|}$ be indexed by $F_k$. For a $k$-decomposition $L$ of $G$, let $ν_{\bf v}(L) = \sum_{F \in F_k} {\bf v}_F d_{L,F}$ where $d_{L,F}$ is the fraction of elements of $L$ isomorphic to $F$. Let $ν_{\bf v}(G) = \max_{L} ν_{\bf v}(L)$ and $ν_{\bf v}(n)=\min\{ν_{\bf v}(G):|V(G)|=n\}$. It is not difficult to prove that the sequence $ν_{\bf v}(n)$ has a limit so let $ν_{\bf v} = \lim_{n \rightarrow \infty} ν_{\bf v}(n)$. Replacing $k$-decompositions with their fractional relaxations, one obtains the (polynomial time computable) fractional analogue $ν_{\bf v}^*(G)$ and corresponding fractional values $ν^*_{\bf v}(n)$ and $ν^*_{\bf v}$. Our first main result is that for each ${\bf v} \in {\mathbb R}^{|F_k|}$ $$ ν_{\bf v} = ν^*_{\bf v}\;. $$ Further, there is a polynomial time algorithm that produces a decomposition $L$ of a $k$-decomposable graph such that $ν_{\bf v}(L) \ge ν_{\bf v} - o_n(1)$. A similar result holds when $F_k$ is the family of all tournaments on $k$ vertices and when $F_k$ is the family of all edge-colorings of $K_k$. We use these results to obtain new and improved bounds on several decomposition results. For example, we prove that every $n$-vertex tournament which is $3$-divisible has a triangle decomposition in which the number of directed triangles is less than $0.0222n^2(1+o(1))$ and that every $5$-decomposable $n$-vertex graph has a $5$-decomposition in which the fraction of cycles of length $5$ is $o_n(1)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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