Minimum Outdegree Research Articles

New results on a problem of Alon

In 2006, Alon proposed a problem of characterizing all four-tuples (n,m,s,d) such that every digraph on n vertices of minimum out-degree at least s contains a subdigraph on m vertices of minimum out-degree at least d. He in particular asked whether there exists an absolute constant c such that every digraph on 2n vertices of minimum out-degree at least s contains a subdigraph on n vertices of minimum out-degree at least s2−c? Recently, Steiner resolved this case in the negative by showing that for arbitrarily large n, there exists a tournament on 2n vertices of minimum out-degree s=n−1, in which the minimum out-degree of every subdigraph on n vertices is at most s2−(12+o(1))log3⁡s.In this paper, we study the above problem and present two new results. The first result is that for arbitrary large n and any integer α≥2, there exists a digraph on αn vertices of minimum out-degree s=n−1 satisfying that the minimum out-degree of every subdigraph on n vertices is at most sα−(1α+o(1))logα+1⁡s. The second result is that for arbitrary large n and any r≥3, there exists a digraph on 2n vertices of girth r and minimum out-degree s satisfying that the minimum out-degree of every subdigraph on n vertices is at most s2−(12+o(1))logr⁡s if r is odd, and is at most s2−(12+o(1))logr+1⁡s if r is even.

Seymour's second neighbourhood conjecture: random graphs and reductions

AbstractA longstanding conjecture of Seymour states that in every oriented graph there is a vertex whose second outneighbourhood is at least as large as its outneighbourhood. In this short note we show that, for any fixed , a.a.s. every orientation of satisfies Seymour's conjecture (as well as a related conjecture of Sullivan). This improves on a recent result of Botler, Moura and Naia. Moreover, we show that is a natural barrier for this problem, in the following sense: for any fixed , Seymour's conjecture is actually equivalent to saying that, with probability bounded away from 0, every orientation of satisfies Seymour's conjecture. This provides a first reduction of the problem. For a second reduction, we consider minimum degrees and show that, if Seymour's conjecture is false, then there must exist arbitrarily large strongly‐connected counterexamples with bounded minimum outdegree. Contrasting this, we show that vertex‐minimal counterexamples must have large minimum outdegree.

Seymour's Second Neighborhood Conjecture for orientations of (pseudo)random graphs

Seymour's Second Neighborhood Conjecture (SNC) states that every oriented graph contains a vertex whose second neighborhood is as large as its first neighborhood. We investigate the SNC for orientations of both binomial and pseudo random graphs, verifying the SNC asymptotically almost surely (a.a.s.)1.for all orientations of G(n,p) if limsupn→∞p<1/4; and2.for a uniformly-random orientation of each weakly (p,Anp)-bijumbled graph of order n and density p, where p=Ω(n−1/2) and 1−p=Ω(n−1/6) and A>0 is a universal constant independent of both n and p. We also show that a.a.s. the SNC holds for almost every orientation of G(n,p). More specifically, we prove that a.a.s.1.for all ε>0 and p=p(n) with limsupn→∞p≤2/3−ε, every orientation of G(n,p) with minimum outdegree Ωε(n) satisfies the SNC; and2.for all p=p(n), a random orientation of G(n,p) satisfies the SNC. We remark that either (iii) or (iv) confirms the SNC for almost every oriented graph.

Two disjoint cycles in digraphs

AbstractBermond and Thomassen conjectured that every digraph with minimum outdegree at least contains vertex disjoint cycles. So far the conjecture was verified for . Here we generalise the question asking for all outdegree sequences which force vertex disjoint cycles and give the full answer for .

Vertex‐disjoint cycles of the same length in tournaments

AbstractA tournament is a digraph which has exactly one arc between any two vertices. Let and be two positive integers with . A ‐cycle is a cycle of length . We prove that for any real number , there exists a constant such that for every , any tournament with minimum outdegree at least contains disjoint ‐cycles. This generalizes a result by Bang‐Jensen, Bessy, and Thomassé for disjoint 3‐cycles. The linear factor of minimum outdegree is the best possible.

Seymour’s Second Neighborhood Conjecture for m-Free Oriented Graphs

Let D = V , E be an oriented graph with minimum out-degree δ + . For x ∈ V D , let d D + x and d D + + x be the out-degree and second out-degree of x in D , respectively. For a directed graph D , we say that a vertex x ∈ V D is a Seymour vertex if d D + + x ≥ d D + x . Seymour in 1990 conjectured that each oriented graph has a Seymour vertex. A directed graph D is called m -free if there are no directed cycles with length at most m in D . A directed graph D = V , E is called k -transitive if, for any directed x y -path of length k , there exists x , y ∈ E . In this paper, we show that (1) each δ + − 2 -free oriented graph has a Seymour vertex and (2) each vertex with minimum out-degree in m -free and 2 m + 2 -transitive oriented graph is a Seymour vertex. The latter result improves a theorem of Daamouch (2021).

Degree Conditions Forcing Directed Cycles

Abstract Caccetta–Häggkvist conjecture is a longstanding open problem on degree conditions that force an oriented graph to contain a directed cycle of a bounded length. Motivated by this conjecture, Kelly, Kühn, and Osthus initiated a study of degree conditions forcing the containment of a directed cycle of a given length. In particular, they found the optimal minimum semidegree, that is, the smaller of the minimum indegree and the minimum outdegree, which forces a large oriented graph to contain a directed cycle of a given length not divisible by $3$, and conjectured the optimal minimum semidegree for all the other cycles except the directed triangle. In this paper, we establish the best possible minimum semidegree that forces a large oriented graph to contain a directed cycle of a given length divisible by $3$ yet not equal to $3$, hence fully resolve the conjecture by Kelly, Kühn, and Osthus. We also find an asymptotically optimal semidegree threshold of any cycle with a given orientation of its edges with the sole exception of a directed triangle.

On judicious bipartitions of directed graphs

Let d≥4 be an fixed integer. Lee, Loh and Sudakov (2016) [17] conjectured that every directed graph D with m arcs and minimum outdegree at least d admits a bipartition V(D)=V1∪V2 such thatmin⁡{e(V1,V2),e(V2,V1)}≥(d−12(2d−1)+o(1))m, where e(Vi,Vj) denote the number of arcs in D from Vi to Vj for {i,j}={1,2}. Let Kd,2→ denote the directed graph obtained by orienting each edge of a bipartite graph K2,d from the part of size d to the other part. In this paper, we show that the conjecture holds for Kd,2→-free directed graphs. Then, we prove that the conjecture holds under the additional condition that the minimum indegree is also at least d−1, which improves a result given by Hou, Ma, Yu and Zhang.

Turán number of 3-free strong digraphs with out-degree restriction

A digraph D is r-free if such D has no directed cycles of length at most r, where r is a positive integer. In 1980, Bermond et al. showed that if D is an r-free strong digraph of order n, then the size of D is at most n−r+22+r−2 and the upper bound is tight, for r≥2. Namely, they determined the Turán number of Cr-free strong digraphs of order n, where Cr={C2,C3,…,Cr} and Ci is a directed cycle of length i∈{2,3,…,r}. Specially, for r=3, the maximum size of 3-free strong digraphs of order n is n−12+1. Let Φn(ξ,γ) be a family of 3-free strong digraphs of order n in which the minimum out-degree is at least ξ and the minimum in-degree is at least γ, where both ξ and γ are positive integers. Let φn(ξ,γ) be the maximum size of digraphs of Φn(ξ,γ), i.e., Turán number of 3-free strong digraphs of order n with out and in-degree restrictions. We denote ϕn(ξ,γ)={D∈Φn(ξ,γ):thesizeofDisequaltoφn(ξ,γ)}. Recently, Chen et al. described ϕn(1,1), i.e., all 3-free strong digraphs of order n with size n−12+1. They also gave the bound of φn(2,1), that is, n−12−2≤φn(2,1)≤n−12. In this paper, we improve the upper bound of φn(2,1) to n−12−1 and we thus get n−12−2≤φn(2,1)≤n−12−1. In addition, we show that φn(2,1)=n−12−1 by constructing two 3-free strong digraphs with minimum out-degree two whose size reaches n−12−1 for n=7,8, and verify φn(2,1)=n−12−2 for n=9. As a consequence, Turán number of 3-free strong digraphs of order n with out-degree at least two, i.e., φn(2,1), is one of n−12−1 and n−12−2.

Strong complete minors in digraphs

AbstractKostochka and Thomason independently showed that any graph with average degree $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_{\!\!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_{\!\!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $\overrightarrow{K}_{\!\!r}$ minor, and any tournament with minimum out-degree $\Omega(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_{\!\!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f\;:\;\mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $\overrightarrow{K}_{\!\!r}$ minor, but such a function exists when considering dichromatic number.

(2K + 1)-Connected Tournaments with Large Minimum Out-Degree are K-Linked

Pokrovskiy conjectured that there is a function f: ℕ → ℕ such that any 2k-strongly-connected tournament with minimum out and in-degree at least f(k) is k-linked. In this paper, we show that any (2k + 1)-strongly-connected tournament with minimum out-degree at least some polynomial in k is k-linked, thus resolving the conjecture up to the additive factor of 1 in the connectivity bound, but without the extra assumption that the minimum in-degree is large. Moreover, we show the condition on high minimum out-degree is necessary by constructing arbitrarily large tournaments that are (2.5k − 1)-strongly-connected tournaments but are not k-linked.

Partitioning digraphs with outdegree at least 4

AbstractScott asked the question of determining such that if is a digraph with arcs and minimum outdegree then has a partition such that , where (respectively, ) is the number of arcs from to (respectively, from to ). Lee, Loh, and Sudakov showed that and , and conjectured that for . In this paper, we show and prove some partial results for .

Majority Colorings of Sparse Digraphs

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that every digraph has a majority $4$-coloring and conjectured that every digraph admits a majority $3$-coloring. They observed that the Local Lemma implies the conjecture for digraphs of large enough minimum out-degree if, crucially, the maximum in-degree is bounded by a(n exponential) function of the minimum out-degree.&#x0D; Our goal in this paper is to develop alternative methods that allow the verification of the conjecture for natural, broad digraph classes, without any restriction on the in-degrees. Among others, we prove the conjecture 1) for digraphs with chromatic number at most $6$ or dichromatic number at most $3$, and thus for all planar digraphs; and 2) for digraphs with maximum out-degree at most $4$. The benchmark case of $r$-regular digraphs remains open for $r \in [5,143]$.&#x0D; Our inductive proofs depend on loaded inductive statements about precoloring extensions of list-colorings. This approach also gives rise to stronger conclusions, involving the choosability version of majority coloring.&#x0D; We also give further evidence towards the existence of majority-$3$-colorings by showing that every digraph has a fractional majority 3.9602-coloring. Moreover we show that every digraph with large enough minimum out-degree has a fractional majority $(2+\varepsilon)$-coloring.

Graph Orientation with Edge Modifications

The goal of an outdegree-constrained edge-modification problem is to find a spanning subgraph or supergraph [Formula: see text] of an input undirected graph [Formula: see text] such that either: (Type I) the number of edges in [Formula: see text] is minimized or maximized and [Formula: see text] can be oriented to satisfy some specified constraints on the vertices’ resulting outdegrees; or: (Type II) among all subgraphs or supergraphs of [Formula: see text] that can be constructed by deleting or inserting a fixed number of edges, [Formula: see text] admits an orientation optimizing some objective involving the vertices’ outdegrees. This paper introduces eight new outdegree-constrained edge-modification problems related to load balancing called (Type I) MIN-DEL-MAX, MIN-INS-MIN, MAX-INS-MAX, and MAX-DEL-MIN and (Type II) [Formula: see text]-DEL-MAX, [Formula: see text]-INS-MIN, [Formula: see text]-INS-MAX, and [Formula: see text]-DEL-MIN. In each of the eight problems, the input is a graph and the goal is to delete or insert edges so that the resulting graph has an orientation in which the maximum outdegree (taken over all vertices) is small or the minimum outdegree is large. We first present a framework that provides algorithms for solving all eight problems in polynomial time on unweighted graphs. Next we investigate the inapproximability of the edge-weighted versions of the problems, and design polynomial-time algorithms for six of the problems on edge-weighted trees.

A decomposition for digraphs with minimum outdegree 3 having no vertex disjoint cycles of different lengths

A decomposition for digraphs with minimum outdegree 3 having no vertex disjoint cycles of different lengths

On bipartitions of directed graphs with small semidegree

Let D be a directed graph. The minimum semidegree of D is defined to be the minimum value of the minimum outdegree and the minimum indegree of D. For nonempty sets S,T⊆V(D), we use e(S,T) to denote the number of arcs in D from S to T. If D has m arcs and positive minimum semidegree, then we show that D admits a bipartition V(D)=V1∪V2 such that min{e(V1,V2),e(V2,V1)}≥(1∕6+o(1))m. We also prove that if the minimum semidegree is at least two (or three, respectively), then the constant can be increased to 1∕5 (or 3∕14, respectively). These partly answer a question of Hou and Wu (2018).

A Note on Min–Max Pair in Tournaments

We prove the following results in this paper. Let T be a tournament of order $$n\ge 3$$ with vertex set V and arc set E, x a vertex of maximum out-degree and y a vertex of minimum out-degree of T. If $$yx\in E$$ then there exists a path of length i from x to y for any i with $$2\le i \le n-1$$ ; and if $$xy\in E$$ , then there exists a path of length i from x to y for any i with $$3\le i \le n-1$$ unless xy is exceptional. We also give a very short discussion to almost regular tournaments and prove a result of Jakobsen.

Rainbow triangles and the Caccetta‐Häggkvist conjecture

AbstractA famous conjecture of Caccetta and Häggkvist is that in a digraph on vertices and minimum outdegree at least n/r there is a directed cycle of length or less. We consider the following generalization: in an undirected graph on vertices, any collection of disjoint sets of edges, each of size at least n/r, has a rainbow cycle of length or less. We focus on the case and prove the existence of a rainbow triangle under somewhat stronger conditions than in the conjecture. In our main result, whenever is larger than a suitable polynomial in , we determine the maximum number of edges in an ‐vertex edge‐colored graph where all color classes have size at most and there is no rainbow triangle. Moreover, we characterize the extremal graphs for this problem.

A bound on judicious bipartitions of directed graphs

Judicious partitioning problems on graphs ask for partitions that bound several quantities simultaneously, which have received a lot of attentions lately. Scott asked the following natural question: What is the maximum constant $c_d$ such that every directed graph $D$ with $m$ arcs and minimum outdegree $d$ admits a bipartition $V(D)= V_1\cup V_2$ satisfying $\min\{e(V_1, V_2), e(V_2, V_1)\}\ge c_d m$? Here, for $i=1,2$, $e(V_{i},V_{3-i})$ denotes the number of arcs in $D$ from $V_{i}$ to $V_{3-i}$. Lee, Loh, and Sudakov %[Judicious partitions of directed graphs, Random Struct. Alg. 48 %(2016) 147--170] conjectured that every directed graph $D$ with $m$ arcs and minimum outdegree at least $d\ge 2$ admits a bipartition $V(D)=V_1\cup V_2$ such that \[ \min\{e(V_1,V_2),e(V_2,V_1)\}\geq \Big(\frac{d-1}{2(2d-1)}+ o(1)\Big)m. \] %While it is not known whether or not the minimum outdegree condition %alone is sufficient, w We show that this conjecture holds under the additional natural condition that the minimum indegree is also at least $d$.

On the Number of Vertex-Disjoint Cycles in Digraphs

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. This conjecture is famous as ...