The sets of flattened partitions with forbidden patterns

Nullspace Vertex Partition in Graphs

The core vertex set of a graph is an invariant of the graph. It consists of those vertices associated with the non-zero entries of the nullspace vectors of a $\{0,1\}$-adjacency matrix. The remaining vertices of the graph form the core--forbidden vertex set. For graphs with independent core vertices, such as bipartite minimal configurations and trees, the nullspace induces a well defined three part vertex partition. The parts of this partition are the core vertex set, their neighbours and the remote core--forbidden vertices. The set of the remote core--forbidden vertices are those not adjacent…

Ramsey-type results for path covers and path partitions. II. Digraphs

Recently, the authors gave Ramsey-type results for the path cover/partition number of graphs. In this paper, we continue the research about them focusing on digraphs, and find a relationship between the path cover/partition number and forbidden structures in digraphs. Let $D$ be a weakly connected digraph. A family $\mathcal{P}$ of subdigraphs of $D$ is called a {\it path cover} (resp. a {\it path partition}) of $D$ if $\bigcup _{P\in \mathcal{P}}V(P)=V(D)$ (resp. $\dot\bigcup _{P\in \mathcal{P}}V(P)=V(D)$) and every element of $\mathcal{P}$ is a directed path. The minimum cardinality of a pat…

Flow-Partitionable Signed Graphs

The NP-hard problem of correlation clustering is to partition a signed graph such that the number of conflicts between the partition and the signature of the graph is minimized. This paper studies graph signatures that allow the optimal partition to be found efficiently. We define the class of flow-partitionable signed graphs, which have the property that the standard linear programming relaxation based on so-called cycle inequalities is tight. In other words, flow-partitionable signed graphs satisfy an exact max-multiflow-min-multicut relation in the associated instances of minimum multicut. …

Separable d-permutations and guillotine partitions

We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between $d$-dimensional permutations and guillotine partitions of a $2^{d-1}$-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain results on a…

Improved bounds on maximum sets of letters in sequences with forbidden alternations

Let $A_{s,k}(m)$ be the maximum number of distinct letters in any sequence which can be partitioned into $m$ contiguous blocks of pairwise distinct letters, has at least $k$ occurrences of every letter, and has no subsequence forming an alternation of length $s$. Nivasch (2010) proved that $A_{5, 2d+1}(m) = θ( m α_{d}(m))$ for all fixed $d \geq 2$. We show that $A_{s+1, s}(m) = \binom{m- \lceil \frac{s}{2} \rceil}{\lfloor \frac{s}{2} \rfloor}$ for all $s \geq 2$, $A_{5, 6}(m) = θ(m \log \log m)$, and $A_{5, 2d+2}(m) = θ(m α_{d}(m))$ for all fixed $d \geq 3$.

Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

Consider the noncrossing set partitions of an $n$-element set which either do not contain the block $\{n-1,n\}$, or which do not contain the singleton block $\{n\}$ whenever $1$ and $n-1$ are in the same block. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and show that it is a supersolvable lattice, and therefore lexicographically shellable. We give a combinatorial model for the NBB bases of this lattice and derive an explicit formula for the value of its Möbius function between least and greatest element. This work is motivated by a rec…

The Chromatic Number of $\mathbb{R}^{n}$ with Multiple Forbidden Distances

Let $A\subset\mathbb{R}_{>0}$ be a finite set of distances, and let $G_{A}(\mathbb{R}^{n})$ be the graph with vertex set $\mathbb{R}^{n}$ and edge set $\{(x,y)\in\mathbb{R}^{n}:\ \|x-y\|_{2}\in A\}$, and let $χ(\mathbb{R}^{n},A)=χ\left(G_{A}(\mathbb{R}^{n})\right)$. Erdős asked about the growth rate of the $m$-distance chromatic number \[ \barχ(\mathbb{R}^{n};m)=\max_{|A|=m}χ(\mathbb{R}^{n},A). \] We improve the best existing lower bound for $\barχ(\mathbb{R}^{n};m)$, and show that \[ \barχ(\mathbb{R}^{n};m)\geq\left(Γ_χ\sqrt{m+1}+o(1)\right)^{n} \] where $Γ_χ=0.79983\dots$ is an explicit cons…

Black-and-white threshold graphs

Let k be a natural number. We introduce k-threshold graphs. We show that there exists an O(n^3) algorithm for the recognition of k-threshold graphs for each natural number k. k-Threshold graphs are characterized by a finite collection of forbidden induced subgraphs. For the case k=2 we characterize the partitioned 2-threshold graphs by forbidden induced subgraphs. We introduce restricted -, and special 2-threshold graphs. We characterize both classes by forbidden induced subgraphs. The restricted 2-threshold graphs coincide with the switching class of threshold graphs. This provides a decompos…

Point partition numbers: perfect graphs

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $t\geq 1$, denote by $\mathcal{MG}_t$ the class of graphs whose maximum multiplicity is at most $t$. A graph $G$ is called strictly $t$-degenerate if every non-empty subgraph $H$ of $G$ contains a vertex $v$ whose degree in $H$ is at most $t-1$. The point partition number $χ_t(G)$ of $G$ is smallest number of colors needed to color the vertices of $G$ so that each vertex receives a color and vertices with the same color induce a strictly $t$-degenerate subgraph of $G$. So $χ_1$ is …

禁止传递竞赛图爆破的有向图与有向图的典型结构

The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments

研究禁止传递竞赛图爆破的有向图与有向图的典型结构，证明几乎所有此类图都具有特定r-划分，并给出计数渐近公式。

混合线性布局中的禁止模式

Forbidden Patterns in Mixed Linear Layouts

本文针对混合线性布局缺乏禁止模式表征的问题，引入类似扭转和彩虹的厚模式，证明有界度图族的混合页数有界当且仅当最大厚模式大小有界，并指出对固定页数k≥2不存在有限禁止模式精确表征。