Combinatorics, Probability and Computing最新文献

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Polarised random -SAT 偏振随机-SAT

Combinatorics, Probability and Computing

Pub Date : 2023-07-20 DOI: 10.1017/s0963548323000226

Joel Larsson Danielsson, Klas Markström

In this paper we study a variation of the random $k$-SAT problem, called polarised random $k$-SAT, which contains both the classical random $k$-SAT model and the random version of monotone $k$-SAT another well-known NP-complete version of SAT. In this model there is a polarisation parameter $p$, and in half of the clauses each variable occurs negated with probability $p$ and pure otherwise, while in the other half the probabilities are interchanged. For $p=1/2$ we get the classical random $k$-SAT model, and at the other extreme we have the fully polarised model where

在本文中，我们研究了随机$k$ -SAT问题的一个变体，称为极化随机$k$ -SAT，它既包含经典的随机$k$ -SAT模型，也包含单调的随机版本$k$ -SAT，另一个著名的np完全版本的SAT。在这个模型中，有一个极化参数$p$，在一半的子句中，每个变量都以概率为负$p$和纯否则。而另一半的概率是互换的。对于$p=1/2$，我们得到经典的随机$k$ -SAT模型，而在另一个极端，我们有完全极化的模型，$p=0$，或1。这里只有两种类型的子句:一种是所有$k$变量纯出现的子句，另一种是所有$k$变量为负值的子句。也就是说，对于$p=0$和$p=1$，我们得到一个随机单调的实例$k$ -SAT。我们表明，满意度阈值不会随着$p$远离$frac{1}{2}$而降低，因此，极化随机$k$ -SAT与$pneq frac{1}{2}$的满意度阈值是随机$k$ -SAT阈值的上界。因此，随机单调$k$ -SAT的可满足阈值至少与随机$k$ -SAT一样大，并且我们推测，对于固定的$k$，这两个阈值渐近地重合。

引用次数: 0

The codegree Turán density of tight cycles minus one edge 馀度Turán紧环的密度减去一条边

Combinatorics, Probability and Computing

Pub Date : 2023-07-05 DOI: 10.1017/s0963548323000196

Simón Piga, Marcelo Sales, Bjarne Schülke

Given $alpha gt 0$ and an integer $ell geq 5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $alpha n$ edges contains a copy of $C_ell ^{-}$, a tight cycle on $ell$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidický.

给定$alpha gt 0$和一个整数$ell geq 5$，我们证明了$n$顶点上每一个足够大的$3$ -均匀超图$H$，其中每两个顶点至少包含$alpha n$条边，其中包含$C_ell ^{-}$的副本，即$ell$顶点上的紧环减去一条边。这改进了先前由Balogh、Clemen和Lidický得出的结果。

引用次数: 1

Unavoidable patterns in locally balanced colourings 不可避免的图案在局部平衡的色彩

Combinatorics, Probability and Computing

Pub Date : 2023-06-01 DOI: 10.1017/s0963548323000160

Nina Kamčev, Alp Müyesser

Which patterns must a two-colouring of

$K_n$

contain if each vertex has at least

$varepsilon n$

red and

$varepsilon n$

blue neighbours? We show that when

$varepsilon gt 1/4$

$K_n$

must contain a complete subgraph on

$Omega (log n)$

vertices where one of the colours forms a balanced complete bipartite graph. When

$varepsilon leq 1/4$

, this statement is no longer true, as evidenced by the following colouring

$chi$

$K_n$

. Divide the vertex set into

如果一个双色的$K_n$顶点至少有$varepsilon n$红色和$varepsilon n$蓝色相邻，那么它必须包含哪些图案?我们证明当$varepsilon gt 1/4$, $K_n$必须包含$Omega (log n)$顶点上的完全子图，其中一个颜色形成平衡的完全二部图。当$varepsilon leq 1/4$时，这种说法不再是正确的，如下所示$K_n$的着色$chi$。将顶点集分成$4$和$V_1,V_2,V_3, V_4$大小几乎相等的部分，并让蓝色类由$(V_1,V_2)$, $(V_2,V_3)$, $(V_3,V_4)$之间的边以及$V_2$和$V_3$内部包含的边组成。令人惊讶的是，我们发现这种障碍在以下意义上是独特的。任何两个着色的$K_n$，其中每个顶点至少有$varepsilon n$红色和$varepsilon n$蓝色的邻居(与$varepsilon gt 0$)包含一个顶点集$S$，其阶为$Omega _{varepsilon }(log n)$，其中一个颜色类形成一个平衡的完全二部图，或者与$chi$具有相同的着色。

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引用次数: 0

The bunkbed conjecture holds in the limit 双层猜想在极限下成立

Combinatorics, Probability and Computing

Pub Date : 2022-12-14 DOI: 10.1017/s096354832200027x

Tom Hutchcroft, Alexander Kent, Petar Nizić-Nikolac

Let $G=(V,E)$ be a countable graph. The Bunkbed graph of $G$ is the product graph $G times K_2$, which has vertex set $Vtimes {0,1}$ with “horizontal” edges inherited from $G$ and additional “vertical” edges connecting $(w,0)$ and $(w,1)$ for each $w in V$. Kasteleyn’s Bunkbed conjecture states that for each $u,v in V$ and

让 $G=(V,E)$ 是一个可数图。的双层图 $G$ 是乘积图 $G times K_2$，它有顶点集 $Vtimes {0,1}$ 的“水平”边缘 $G$ 和额外的“垂直”边连接 $(w,0)$ 和 $(w,1)$ 对于每一个 $w in V$． Kasteleyn的双层床猜想指出，对于每一个 $u,v in V$ 和 $pin [0,1]$，顶点 $(u,0)$ 至少有可能被连接到 $(v,0)$ 至于 $(v,1)$ 在伯努利下$p$ 双层图上的键渗透。我们证明了这个猜想在 $p uparrow 1$ 极限是指对于每一个有限图 $G$ 存在 $varepsilon (G)gt 0$ 这样一来，床铺猜想就成立了 $p geqslant 1-varepsilon (G)$．

引用次数: 0

A bipartite version of the Erdős–McKay conjecture Erdős-McKay猜想的二分版本

Combinatorics, Probability and Computing

Pub Date : 2022-12-09 DOI: 10.1017/s0963548322000347

Eoin Long, Laurenţiu Ploscaru

An old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(!log n)$ then the graph contains induced subgraphs of each size from ${0,1,ldots, Omega big(n^2big)}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n times n$ bipartite graph are of order $O(!log n)$, then the graph contains induced subgraphs of each size from ${0,1,ldots, Omega big(n^2big)}$.

Erdős和McKay的一个老猜想指出，如果一个$n$顶点图中的所有齐次集合都是$O(!log n)$阶的，那么这个图包含来自${0,1,ldots, Omega big(n^2big)}$的各种大小的诱导子图。我们证明了这个猜想的一个二部类比:如果一个$n times n$二部图中所有的平衡齐次集合都是$O(!log n)$阶的，那么这个图包含来自${0,1,ldots, Omega big(n^2big)}$的每个大小的诱导子图。

引用次数: 0