弱半正则剖分的数学复杂性与表示数
Algorithmic Complexity of Weakly Semiregular Partitioning and the Representation Number
摘要：
一个图$G$是{ \it弱半正则}，如果有两个数$a,b$，使得每个顶点的度是$a$或$b$。图$G$的{ \it弱半正则数}，记为$wr(G)$，是$G$的边集可划分成的子集的最小数目，使得每个子集所导出的子图是弱半正则图.本文给出了一个判定给定树的弱半正则数是否为2的多项式时间算法。另一方面，我们证明了，确定是否$ wr(G) = 2 $为一个给定的二部图$ G $在其度集最多有三个号码是{ \bf NP}-完全。在其他结果中，对于每棵树$T$，我们表明，$wr(T)\leq 2\log_2 Δ(T) + \mathcal{O}(1)$，其中$Δ(T)$表示$T$的最大程度。在第二部分的工作中，我们考虑的表示数。一个图$G$有{ \it表示模$r$ }，如果存在一个内射映射$\ell: V (G) \rightarrow \mathbb{Z}_r$使得顶点$v$和$u$相邻当且仅当$|\ell(u) -\ell(v)|$与$r$互质.由$rep(G)$表示的{ \it表示数}是最小的$r$，使得$G$具有以$r$为模的表示。Narayan和Urick指出，对于任意图$G$，确定$rep (G)$是一个困难的问题\cite{narayan2007representations} .在这项工作中，我们证实了这一猜想，并表明，如果$\mathbf{NP\neq P}$，那么对于任何$ε>0$，有没有多项式时间的$(1-ε)\frac{n}{2}$ -近似算法计算的表示数的正则图的$n$顶点。
英文摘要：
A graph $G$ is {\it weakly semiregular} if there are two numbers $a,b$, such that the degree of every vertex is $a$ or $b$. The {\it weakly semiregular number} of a graph $G$, denoted by $wr(G)$, is the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether $ wr(G) = 2 $ for a given bipartite graph $ G $ with at most three numbers in its degree set is {\bf NP}-complete. Among other results, for every tree $T$, we show that $wr(T)\leq 2\log_2 Δ(T) + \mathcal{O}(1)$, where $Δ(T)$ denotes the maximum degree of $T$. In the second part of the work, we consider the representation number. A graph $G$ has a {\it representation modulo $r$} if there exists an injective map $\ell: V (G) \rightarrow \mathbb{Z}_r$ such that vertices $v$ and $u$ are adjacent if and only if $|\ell(u) -\ell(v)|$ is relatively prime to $r$. The {\it representation number}, denoted by $rep(G)$, is the smallest $r$ such that $G$ has a representation modulo $r$. Narayan and Urick conjectured that the determination of $rep (G)$ for an arbitrary graph $G$ is a difficult problem \cite{narayan2007representations}. In this work, we confirm this conjecture and show that if $\mathbf{NP\neq P}$, then for any $ε>0$, there is no polynomial time $(1-ε)\frac{n}{2}$-approximation algorithm for the computation of representation number of regular graphs with $n$ vertices.
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