scholarly journals Subdivisions in Digraphs of Large Out-Degree or Large Dichromatic Number

10.37236/6521 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Pierre Aboulker ◽  
Nathann Cohen ◽  
Frédéric Havet ◽  
William Lochet ◽  
Phablo F. S. Moura ◽  
...  
Keyword(s):  
Directed Path ◽  
Directed Paths ◽  
Oriented Path ◽  
Transitive Tournament ◽  
Dichromatic Number

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the existence of $f(5)$ remains unknown. In this paper, we show that if $D$ is an oriented path, or an in-arborescence (i.e., a tree with all edges oriented towards the root) or the union of two directed paths from $x$ to $y$ and a directed path from $y$ to $x$, then every digraph with minimum out-degree large enough contains a subdivision of $D$. Additionally, we study Mader's conjecture considering another graph parameter. The dichromatic number of a digraph $D$ is the smallest integer $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision. We show that any digraph with dichromatic number greater than $4^m (n-1)$ contains every digraph with $n$ vertices and $m$ arcs as a subdivision.

scholarly journals Duality Pairs and Homomorphisms to Oriented and Unoriented Cycles

10.37236/9747 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Santiago Guzmán-Pro ◽  
César Hernández-Cruz
Keyword(s):  
Undirected Graph ◽  
Linear Size ◽  
Directed Path ◽  
Duality Pairs ◽  
Finite Set ◽  
Oriented Path ◽  
Transitive Tournament

 In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not \to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a duality pair. As a consequence we obtain that there is a finite set, $F_C$, such that an undirected graph is homomorphic to $C$, if and only if it admits an $F_C$-free orientation. As a byproduct of the proposed duality pairs, we show that if $T$ is an oriented tree of height at most $3$, one can choose a dual of $T$ of linear size with respect to the size of $T$.

scholarly journals Extension of Gyárfás-Sumner Conjecture to Digraphs

10.37236/9906 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Pierre Aboulker ◽  
Pierre Charbit ◽  
Reza Naserasr
Keyword(s):  
Chromatic Number ◽  
Directed Path ◽  
Complete Multipartite Graph ◽  
Color Class ◽  
Acyclic Digraph ◽  
Minimum Number ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Dichromatic Number

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices  in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization  of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph.  Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

Maximum weight induced multicliques and complete multipartite subgraphs in directed path overlap graphs

2015 ◽  
Vol 07 (04) ◽  
pp. 1550061
Author(s):  
Fanica Gavril
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Connected Components ◽  
Directed Path ◽  
Maximum Weight ◽  
Directed Tree ◽  
Directed Paths ◽  
Multipartite Graph ◽  
Overlap Graphs ◽  
Complete Multipartite

A graph is a directed path overlap graph if it is the overlap graph of family of directed paths in a rooted directed tree. A graph is a multiclique if its connected components are cliques. A graph is a complete multipartite graph if it is the complement of a multiclique. A graph is a multiclique-multipartite graph if its vertex set has a partition [Formula: see text], [Formula: see text] such that [Formula: see text] is complete multipartite, [Formula: see text] is a multiclique and every two vertices [Formula: see text], [Formula: see text] are adjacent. We describe a polynomial time algorithm to find a maximum weight induced complete multipartite (MWICM) subgraph in a directed path overlap graph. In addition, we describe polynomial time algorithms to find maximum weight induced (restricted) multicliques (MWIM) and multiclique-multipartite (MWIMM) subgraphs in directed path overlap graphs.

scholarly journals Turán theorems for unavoidable patterns

2021 ◽  
pp. 1-20
Author(s):  
ANTÓNIO GIRÃO ◽  
BHARGAV NARAYANAN
Keyword(s):  
Complete Graph ◽  
Blow Up ◽  
Ramsey Numbers ◽  
Order Of Magnitude ◽  
Transitive Tournament ◽  
Unavoidable Patterns

Abstract We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

scholarly journals Controllability of Weighted and Directed Networks with Nonidentical Node Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Linying Xiang ◽  
Jonathan J. H. Zhu ◽  
Fei Chen ◽  
Guanrong Chen
Keyword(s):  
Graph Theory ◽  
Control Theory ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Directed Path ◽  
Directed Networks ◽  
Leaders And Followers ◽  
Heterogenous Network

The concept of controllability from control theory is applied to weighted and directed networks with heterogenous linear or linearized node dynamics subject to exogenous inputs, where the nodes are grouped into leaders and followers. Under this framework, the controllability of the controlled network can be decomposed into two independent problems: the controllability of the isolated leader subsystem and the controllability of the extended follower subsystem. Some necessary and/or sufficient conditions for the controllability of the leader-follower network are derived based on matrix theory and graph theory. In particular, it is shown that a single-leader network is controllable if it is a directed path or cycle, but it is uncontrollable for a complete digraph or a star digraph in general. Furthermore, some approaches to improving the controllability of a heterogenous network are presented. Some simulation examples are given for illustration and verification.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Linial's Dual Conjecture for Path-Spine Digraphs

2020 ◽  
Author(s):  
Vinícius De Souza Carvalho ◽  
Cândida Nunes Da Silva ◽  
Orlando Lee
Keyword(s):  
Stable Sets ◽  
Hamilton Path ◽  
Directed Paths ◽  
Vertex Disjoint ◽  
Single Path

 Given a digraph D, a coloring 𝒞 of D is a partition of V(D) into stable sets. The k-norm of 𝒞 is defined as ΣC∈𝒞 min{|C|, k}. A coloring of D with minimum k-norm has its k-norm noted by χk(D). A (path)-k-pack of a digraph D is a set of k vertex-disjoint (directed) paths of D. The weight of a k-pack is the number of vertices covered by the k-pack. We denote by λk(D) the weight of a maximum k-pack. Linial conjectured that χk(D) ≤ λk(D) for every digraph. Such conjecture remains open, but has been proved for some classes of digraphs. We prove the conjecture for path-spine digraphs, defined as follows. A digraph D is path-spine if there exists a partition {X, Y} of V(D) such that D[X] has a Hamilton path and every arc in D[Y] belongs to a single path Q. 

scholarly journals Fully Dynamic Transitive Closure in Plane Dags with one Source and one Sink

1994 ◽  
Vol 1 (30) ◽  
Author(s):  
Thore Husfeldt
Keyword(s):  
Transitive Closure ◽  
Upper Bounds ◽  
Directed Acyclic Graphs ◽  
Closure Problem ◽  
Directed Path ◽  
Lower And Upper Bounds ◽  
New Techniques ◽  
Time Dynamic ◽  
Acyclic Graphs ◽  
Closure Algorithm

We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from u to v ?' for arbitrary vertices u and v can be answered in logarithmic time. The size of the data structure and the initialisation time are linear in the number of edges.<br /> <br />The result enlarges the class of graphs for which a logarithmic (or even polylogarithmic) time dynamic transitive closure algorithm exists. Previously, the only algorithms within the stated resource bounds put restrictions on the topology of the graph or on the delete operation. To obtain our result, we use a new characterisation of the transitive closure in plane graphs with one source and one sink and introduce new techniques to exploit this characterisation.<br /> <br />We also give a lower bound of Omega(log n/log log n) on the amortised complexity of the problem in the cell probe model with logarithmic word size. This is the first dynamic directed graph problem with almost matching lower and upper bounds.

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