scholarly journals On Directed Versions of the Hajnal–Szemerédi Theorem

2015 ◽  
Vol 24 (6) ◽  
pp. 873-928 ◽  
Author(s):  
ANDREW TREGLOWN
Keyword(s):  
Directed Graph ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Perfect Matchings ◽  
Large Order ◽  
Vertex Disjoint ◽  
Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

scholarly journals Semi-Degree Threshold for Anti-Directed Hamiltonian Cycles

10.37236/3610 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Louis DeBiasio ◽  
Theodore Molla
Keyword(s):  
Directed Graph ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Directed Graphs ◽  
Hamiltonian Cycles ◽  
Alternate Direction ◽  
Degree Conditions

In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$, then $D$ contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph $D$ to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even $n$, if $D$ is a directed graph on $n$ vertices with minimum out-degree and in-degree at least $\frac{n}{2}+1$, then $D$ contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that $\frac{n}{2}$ is sufficient unless $D$ is one of two counterexamples. This result is sharp.

scholarly journals TILING DIRECTED GRAPHS WITH TOURNAMENTS

2018 ◽  
Vol 6 ◽  
Author(s):  
ANDRZEJ CZYGRINOW ◽  
LOUIS DEBIASIO ◽  
THEODORE MOLLA ◽  
ANDREW TREGLOWN
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Complete Graph ◽  
Directed Graphs ◽  
Type Result ◽  
Vertex Disjoint

The Hajnal–Szemerédi theorem states that for any positive integer $r$ and any multiple $n$ of $r$, if $G$ is a graph on $n$ vertices and $\unicode[STIX]{x1D6FF}(G)\geqslant (1-1/r)n$, then $G$ can be partitioned into $n/r$ vertex-disjoint copies of the complete graph on $r$ vertices. We prove a very general analogue of this result for directed graphs: for any positive integer $r$ with $r\neq 3$ and any sufficiently large multiple $n$ of $r$, if $G$ is a directed graph on $n$ vertices and every vertex is incident to at least $2(1-1/r)n-1$ directed edges, then $G$ can be partitioned into $n/r$ vertex-disjoint subgraphs of size $r$ each of which contain every tournament on $r$ vertices (the case $r=3$ is different and was handled previously). In fact, this result is a consequence of a tiling result for standard multigraphs (that is multigraphs where there are at most two edges between any pair of vertices). A related Turán-type result is also proven.

scholarly journals Disjoint Cycles of Different Lengths in Graphs and Digraphs

10.37236/6921 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Julien Bensmail ◽  
Ararat Harutyunyan ◽  
Ngoc Khang Le ◽  
Binlong Li ◽  
Nicolas Lichiardopol
Keyword(s):  
Minimum Degree ◽  
Directed Graphs ◽  
Probabilistic Method ◽  
Undirected Graphs ◽  
Disjoint Cycles ◽  
Vertex Disjoint ◽  
Degree Bound ◽  
Directed Cycles

In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider other cases such as when the graph is triangle-free, or the $k$ cycles are required to have different lengths modulo some value $r$. In the context of directed graphs, we consider a conjecture of Lichiardopol concerning the least minimum out-degree required for a digraph to have $k$ vertex-disjoint directed cycles of different lengths. We verify this conjecture for tournaments, and, by using the probabilistic method, for some regular digraphs and digraphs of small order.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals The Diameter of Almost Eulerian Digraphs

10.37236/429 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dankelmann ◽  
L. Volkmann
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Upper Bound ◽  
Minimum Degree ◽  
Additive Constant ◽  
Undirected Graphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

scholarly journals Convert a Strongly Connected Directed Graph to a Black-and-White 3-SAT Problem by the Balatonboglár Model

2020 ◽  
Author(s):  
Gábor Kusper ◽  
Csaba Biró
Keyword(s):  
Directed Graph ◽  
Directed Graphs ◽  
Deep Insight ◽  
Communication Graph ◽  
Sat Problem ◽  
Weak Model ◽  
Know How ◽  
Black And White ◽  
Communication Graphs ◽  
Strongly Connected

In a previous paper we defined the Black-and-White SAT problem which has exactly two solutions, where each variable is either true or false. We showed that Black-and-White $2$-SAT problems represent strongly connected directed graphs. We presented also the strong model of communication graphs. In this work we introduce two new models, the weak model, and the Balatonbogl\'{a}r model of communication graphs. A communication graph is a directed graph, where no self loops are allowed. In this work we show that the weak model of a strongly connected communication graph is a Black-and-White SAT problem. We prove a powerful theorem, the so called Transitions Theorem. This theorem states that for any model which is between the strong and the weak model, we have that this model represents strongly connected communication graphs as Blask-and-White SAT problems. We show that the Balatonbogl\'{a}r model is between the strong and the weak model, and it generates $3$-SAT problems, so the Balatonbogl\'{a}r model represents strongly connected communication graphs as Black-and-White $3$-SAT problems. Our motivation to study these models is the following: The strong model generates a $2$-SAT problem from the input directed graph, so it does not give us a deep insight how to convert a general SAT problem into a directed graph. The weak model generates huge models, because it represents all cycles, even non-simple cycles, of the input directed graph. We need something between them to gain more experience. From the Balatonbogl\'{a}r model we learned that it is enough to have a subset of a clause, which represents a cycle in the weak model, to make the Balatonbogl\'{a}r model more compact. We still do not know how to represent a SAT problem as a directed graph, but this work gives a strong link between two prominent fields of formal methods: SAT problem and directed graphs.

scholarly journals Active Semantic Relations in Layered Enterprise Architecture Development

2021 ◽  
pp. 3-16
Author(s):  
Matt Baxter ◽  
Simon Polovina ◽  
Wim Laurier ◽  
Mark von Rosing
Keyword(s):  
Directed Graph ◽  
Enterprise Architecture ◽  
Directed Graphs ◽  
Building Blocks ◽  
Semantic Relations ◽  
Formal Concept ◽  
General Applicability ◽  
Conceptual Graph ◽  
Architecture Development

AbstractEnterprise Architecture (EA) metamodels align an organisation’s business, information and technology resources so that these assets best meet the organisation’s purpose. The Layered EA Development (LEAD) Ontology enhances EA practices by a metamodel with layered metaobjects as its building blocks interconnected by semantic relations. Each metaobject connects to another metaobject by two semantic relations in opposing directions, thus highlighting how each metaobject views other metaobjects from its perspective. While the resulting two directed graphs reveal all the multiple pathways in the metamodel, more desirable would be to have one directed graph that focusses on the dependencies in the pathways. Towards this aim, using CG-FCA (where CG refers to Conceptual Graph and FCA to Formal Concept Analysis) and a LEAD case study, we determine an algorithm that elicits the active as opposed to the passive semantic relations between the metaobjects resulting in one directed graph metamodel. We also identified the general applicability of our algorithm to any metamodel that consists of triples of objects with active and passive relations.

scholarly journals Parameterized Complexity of Directed Spanner Problems

Algorithmica ◽  
2021 ◽  
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
William Lochet ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Directed Graph ◽  
Parameterized Complexity ◽  
Directed Graphs ◽  
Directed Acyclic Graphs ◽  
Polynomial Kernel ◽  
Distortion Parameter ◽  
Spanning Subgraph ◽  
Acyclic Graphs ◽  
Positive Integers ◽  
Additive Distortion

AbstractWe initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

scholarly journals On Komlós’ tiling theorem in random graphs

2019 ◽  
Vol 29 (1) ◽  
pp. 113-127
Author(s):  
Rajko Nenadov ◽  
Nemanja Škorić
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Upper Bound ◽  
Minimum Degree ◽  
Arbitrary Graph ◽  
Spanning Subgraph ◽  
Delta G ◽  
Image Position ◽  
Vertex Disjoint

AbstractGiven graphs G and H, a family of vertex-disjoint copies of H in G is called an H-tiling. Conlon, Gowers, Samotij and Schacht showed that for a given graph H and a constant γ>0, there exists C>0 such that if $p \ge C{n^{ - 1/{m_2}(H)}}$ , then asymptotically almost surely every spanning subgraph G of the random graph 𝒢(n, p) with minimum degree at least $\delta (G) \ge (1 - \frac{1}{{{\chi _{{\rm{cr}}}}(H)}} + \gamma )np$ contains an H-tiling that covers all but at most γn vertices. Here, χcr(H) denotes the critical chromatic number, a parameter introduced by Komlós, and m2(H) is the 2-density of H. We show that this theorem can be bootstrapped to obtain an H-tiling covering all but at most $\gamma {(C/p)^{{m_2}(H)}}$ vertices, which is strictly smaller when $p \ge C{n^{ - 1/{m_2}(H)}}$ . In the case where H = K3, this answers the question of Balogh, Lee and Samotij. Furthermore, for an arbitrary graph H we give an upper bound on p for which some leftover is unavoidable and a bound on the size of a largest H -tiling for p below this value.

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