arXiv 1808.062932018-08math.CO

关于顶点子集的Seymour第二邻域猜想

Seymour's Second Neighborhood Conjecture for Subsets of Vertices

Tyler Seacrest

摘要：

Seymour证明了每个有向简单图都包含一个顶点，该顶点的第二个邻域至少与第一个邻域一样大。在本文中，我们提出了一个猜想，我们证明实际上是等价的：每一个有向简单图包含一个子集的顶点$S$的第二个邻域至少是一样大的第一个。这个子集的观点给了一些洞察原来的猜想。例如，如果有一个反例的第二个邻居猜想的最小程度$δ$，那么存在一个反例在最多${δ+ 1 \choose 2}$顶点。给定顶点$v$，令$d_1^+(v)$和$d_2^+(v)$分别为其第一和第二邻域的大小。一个有向图是$m$自由图，如果$m$上没有有向圈或顶点数较少.设$λ_m$是使得每个无$m$图都含有一个顶点$v$且$d_2^+(v) \geq λ_m d_1^+(v)$为$d_2^+(v) \geq λ_m d_1^+(v)$的最大值.第二个邻域猜想意味着对所有$m \geq 2$都有$λ_m = 1$。Liang和Xu提供了所有$λ_m$的下界，并表明$λ_m \to 1$为$m \to \infty$。我们改进梁和徐的界$m \geq 3$使用这个子集的角度。

英文摘要：

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. In this note, we put forward a conjecture that we prove is actually equivalent: every oriented simple graph contains a subset of vertices $S$ whose second neighborhood is at least as large as its first. This subset perspective gives some insight into the original conjecture. For example, if there is a counterexample to the second neighborhood conjecture with minimum degree $δ$, then there exists a counterexample on at most ${δ+ 1 \choose 2}$ vertices. Given a vertex $v$, let $d_1^+(v)$ and $d_2^+(v)$ be the size of its first and second neighborhoods respectively. A digraph is $m$-free if there is no directed cycle on $m$ or fewer vertices. Let $λ_m$ be the largest value such that every $m$-free graph contains a vertex $v$ with $d_2^+(v) \geq λ_m d_1^+(v)$. The second neighborhood conjecture implies $λ_m = 1$ for all $m \geq 2$. Liang and Xu provided lower bounds for all $λ_m$, and showed that $λ_m \to 1$ as $m \to \infty$. We improve on Liang and Xu's bound for $m \geq 3$ using this subset perspective.

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