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arXiv 1702.06203math.CO
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生成树与小度数生成闭游动

Spanning trees and spanning closed walks with small degrees

Morteza Hasanvand

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22 pages: Some removed parts of former versions will be published in some new papers
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摘要：

设$G$是一个图，$f$是$V(G)$上的正整数值函数.在本文中，我们证明了，如果对所有的$S\subseteq V(G)$，$ω(G\setminus S)<\sum_{v\in S}(f(v)-2)+2+ω(G[S])$，那么$G$有一个生成树$T$包含一个任意给定的匹配，使得对于每个顶点$v$，$d_T(v)\le f(v)$，其中$ω(G\setminus S)$表示$G\setminus S$的分支数，$ω(G[S])$表示导出子图$G[S]$的分支数与顶点集$S$。这是几个结果的改进。其次，证明了如果对所有的$S\subseteq V(G)$，$ω(G\setminus S)\le \sum_{v\in S} (f(v)-1)+1$，$G$允许一个生成闭行走通过任意给定匹配的边，满足每个顶点$v$至多$f(v)$次.这一结果解决了Jackson和Wormald（1990）长期存在的猜想。

英文摘要：

Let $G$ be a graph and let $f$ be a positive integer-valued function on $V(G)$. In this paper, we show that if for all $S\subseteq V(G)$, $ω(G\setminus S)<\sum_{v\in S}(f(v)-2)+2+ω(G[S])$, then $G$ has a spanning tree $T$ containing an arbitrary given matching such that for each vertex $v$, $d_T(v)\le f(v)$, where $ω(G\setminus S)$ denotes the number of components of $G\setminus S$ and $ω(G[S])$ denotes the number of components of the induced subgraph $G[S]$ with the vertex set $S$. This is an improvement of several results. Next, we prove that if for all $S\subseteq V(G)$, $ω(G\setminus S)\le \sum_{v\in S} (f(v)-1)+1$, then $G$ admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex $v$ at most $f(v)$ times. This result solves a long-standing conjecture due to Jackson and Wormald (1990).

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