Minimum Degree Condition Research Articles

Minimum degree conditions for containing an [formula omitted]-regular [formula omitted]-connected spanning subgraph

We study optimal minimum degree conditions when an n-vertex graph G contains an r-regular r-connected spanning subgraph. We prove for r fixed and n large the condition to be δ(G)≥n+r−22 when nr≡0(mod2). This answers a question of M. Kriesell.

A robust Corrádi–Hajnal theorem

AbstractFor a graph and , we denote by the random sparsification of obtained by keeping each edge of independently, with probability . We show that there exists a such that if and is an ‐vertex graph with and , then with high probability contains a triangle factor. Both the minimum degree condition and the probability condition, up to the choice of , are tight. Our result can be viewed as a common strengthening of the seminal theorems of Corrádi and Hajnal, which deals with the extremal minimum degree condition for containing triangle factors (corresponding to in our result), and Johansson, Kahn and Vu, which deals with the threshold for the appearance of a triangle factor in (corresponding to in our result). It also implies a lower bound on the number of triangle factors in graphs with minimum degree at least which gets close to the truth.

Sharp minimum degree conditions for disjoint doubly chorded cycles

Sharp minimum degree conditions for disjoint doubly chorded cycles

On oriented cycles in randomly perturbed digraphs

AbstractIn 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha \gt 0$ , there exists a constant $C$ such that for every $n$ -vertex digraph of minimum semi-degree at least $\alpha n$ , if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$ . Our proofs make use of a variant of an absorbing method of Montgomery.

A Ramsey–Turán theory for tilings in graphs

AbstractFor a ‐vertex graph and an ‐vertex graph , an ‐tiling in is a collection of vertex‐disjoint copies of in . For , the ‐independence number of , denoted , is the largest size of a ‐free set of vertices in . In this article, we discuss Ramsey–Turán‐type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal ‐tilings. Our results unify and generalise previous results of Balogh–Molla–Sharifzadeh [Random Struct. Algoritm. 49 (2016), no. 4, 669–693], Nenadov–Pehova [SIAM J. Discret. Math. 34 (2020), no. 2, 1001–1010] and Balogh–McDowell–Molla–Mycroft [Comb. Probab. Comput. 27 (2018), no. 4, 449–474] on the subject.

A pair degree condition for Hamiltonian cycles in 3-uniform hypergraphs

AbstractWe prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$ -uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$ -uniform hypergraphs by proving a $3$ -uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.

Clique-factors in graphs with sublinear -independence number

Abstract Given a graph $G$ and an integer $\ell \ge 2$ , we denote by $\alpha _{\ell }(G)$ the maximum size of a $K_{\ell }$ -free subset of vertices in $V(G)$ . A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$ -vertex graphs $G$ with $\alpha _{\ell }(G) = o(n)$ , which can be seen as a Ramsey–Turán variant of the celebrated Hajnal–Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$ -factors in $n$ -vertex graphs $G$ with $\alpha _\ell (G)=n^{1-o(1)}$ for all $r\ge \ell \ge 2$ .

Embedding clique-factors in graphs with low ℓ-independence number

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: given μ>0 and integers ℓ,r and n with n∈rN, is it true that there exists an α>0 such that every n-vertex graph G with δ(G)≥max⁡{12,r−ℓr}n+μn and αℓ(G)≤αn contains a Kr-factor? We give a negative answer to this question for the case ℓ≥3r4 by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers r,ℓ with r>ℓ≥34r and μ>0, there exist α>0 and N such that for every n∈rN with n>N, every n-vertex graph G with δ(G)≥(12−ϱℓ(r−1)+μ)n and αℓ(G)≤αn contains a Kr-factor. Here ϱℓ(r−1) is the Ramsey–Turán density for Kr−1 under the ℓ-independence number condition.

Remarks on component factors in K1,r-free graphs

An ℱ-factor is a spanning subgraph H such that each connected component of H is isomorphic to some graph in ℱ. We use Pk and K1,r to denote the path of order k and the star of order r + 1, respectively. In particular, H is called a {P2, P3}-factor of G if ℱ = {P2, P3}; H is called a P≥k-factor of G if ℱ = {Pk, Pk+1,…}, where k ≥ 2; H is called an Sn-factor of G if ℱ = {P2, P3, K1,3,…, K1,n}, where n ≥ 2. A graph G is called a ℱ≥k-factor covered graph if there is a ℱ≥k-factor of G including e for any e ∈ E(G). We call a graph G is K1,r-free if G does not contain an induced subgraph isomorphic to K1,r. In this paper, we give a minimum degree condition for the K1,r-free graph with an Sn-factor and the K1,r-free graph with a ℱ≥3-factor, respectively. Further, we obtain sufficient conditions for K1,r-free graphs to be ℱ≥2-factor, ℱ≥3-factor or {P2, P3}-factor covered graphs. In addition, examples show that our results are sharp.

Minimum degree conditions for the existence of a sequence of cycles whose lengths differ by one or two

AbstractGao, Huo, Liu, and Ma proved a result on the existence of paths connecting specified two vertices whose lengths differ by one or two. By using this result, they settled two famous conjectures due to Thomassen. In this paper, we improve their result, and obtain a generalization of a result of Bondy and Vince.

Tiling multipartite hypergraphs in quasi-random hypergraphs

Given k≥2 and two k-graphs (k-uniform hypergraphs) F and H, an F-factor in H is a set of vertex disjoint copies of F that together covers the vertex set of H. Lenz and Mubayi studied the F-factor problems in quasi-random k-graphs with minimum degree Ω(nk−1). In particular, they constructed a sequence of 1/8-dense quasi-random 3-graphs H(n) with minimum degree Ω(n2) and minimum codegree Ω(n) but with no K2,2,2-factor. We prove that if p>1/8 and F is a 3-partite 3-graph with f vertices, then for sufficiently large n, all p-dense quasi-random 3-graphs of order n with minimum codegree Ω(n) and f|n have F-factors. That is, 1/8 is the density threshold for ensuring all 3-partite 3-graphs F-factors in quasi-random 3-graphs given a minimum codegree condition Ω(n). Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any p∈(0,1) and n≥n0, there exist p-dense quasi-random 3-graphs of order n with minimum degree Ω(n2) having no K2,2,2-factor. In particular, we study the optimal density threshold of F-factors for each 3-partite 3-graph F in quasi-random 3-graphs given a minimum codegree condition Ω(n).In addition, we also study F-factor problems for k-partite k-graphs F with stronger quasi-random assumption and minimum degree Ω(nk−1).

Minimum degree condition of Berge Hamiltonicity in random 3-uniform hypergraphs

A graph H has Hamiltonicity if it contains a cycle which covers each vertex of H. In graph theory, Hamiltonicity is a classical and worth studying problem. In 1952, Dirac proved that any n-vertex graph H with minimum degree at least ?n/2? has Hamiltonicity. In 2012, Lee and Sudakov proved that if p ? logn/n , then asympotically almost surely each n-vertex subgraph of random graph G(n,p) with minimum degree at least (1/2 + o(1))np has Hamiltonicity. In this paper, we exend Dirac?s theorem to random 3-uniform hypergraphs. The random 3-uniform hypergraph model H3(n, p) consists of all 3-uniform hypergraphs on n vertices and every possible edge appears with probability p randomly and independently. We prove that if p ? logn/n2, then asympotically almost surely every n-vertex subgraph of H3(n, p) with minimum degree at least (1/4 + o(1))(n 2)p has Berge Hamiltonicity. The value logn/n2 and constant 1/4 both are best possible.

Minimum Degrees for Powers of Paths and Cycles

We study minimum degree conditions under which a graph $G$ contains $k^{th}$ power of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends a result of Allen, Bottcher and Hladký concerning the containment of squares of paths and squares of cycles of arbitrary specified lengths and settles a conjecture of theirs in the affirmative.

Shadows of 3-Uniform Hypergraphs under a Minimum Degree Condition

We prove a minimum degree version of the Kruskal--Katona theorem for triple systems: given $d\ge 1/4$ and a triple system $\mathcal{F}$ on $n$ vertices with minimum degree $\delta(\mathcal{F})\ge d\binom n2$, we obtain asymptotically tight lower bounds for the size of its shadow. Equivalently, for $t\ge n/2-1$, we asymptotically determine the minimum size of a graph on $n$ vertices, in which every vertex is contained in at least $\binom t2$ triangles. This can be viewed as a variant of the Rademacher--Turán problem.

Minimum degree conditions for the strength and bandwidth of graphs

A numbering f of a graph G is a labeling that assigns distinct elements of the set 1 , 2 , … , V G to the vertices of G . The strength str f G of a numbering f : V G → 1 , 2 , … , V G of G is defined by str f G = max f u + f v u v ∈ E G , and the strength str G of a graph G is str G = min str f G f is a numbering of G . In this paper, we investigate minimum degree conditions for the strength of graphs. From the study conducted in this paper, certain degree sequences of graphs naturally arise and we provide a proof of the fact that these degree sequences determine a unique graph realization. In addition, we establish a parallel bandwidth result to the one on strength of graphs. Furthermore, we enlarge the class of k -stable properties known so far. These results follow naturally from the work conducted in this paper.

Minimum degree and the graph removal lemma

AbstractThe clique removal lemma says that for every and , there exists some so that every ‐vertex graph with fewer than copies of can be made ‐free by removing at most edges. The dependence of on in this result is notoriously difficult to determine: it is known that must be at least super‐polynomial in , and that it is at most of tower type in . We prove that if one imposes an appropriate minimum degree condition on , then one can actually take to be a linear function of in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super‐polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.

A note on minimum degree condition for Hamilton (a,b)-cycles in hypergraphs

Let k,a,b be positive integers with a+b=k. A k-uniform hypergraph is called an (a,b)-cycle if there is a partition (A0,B0,A1,B1,…,At−1,Bt−1) of the vertex set with |Ai|=a, |Bi|=b such that Ai∪Bi and Bi∪Ai+1 (subscripts module t) are edges for all i=0,1,…,t−1. Let H be a k-uniform n-vertex hypergraph with n≥5k and n divisible by k. By applying the concentration inequality for intersections of a uniform hypergraph with a random matching developed by Frankl and Kupavskii, we show that if there exists α∈(0,1) such that δa(H)≥(α+o(1))(n−ab) and δb(H)≥(1−α+o(1))(n−ba), then H contains a Hamilton (a,b)-cycle. As a corollary, we prove that if δℓ(H)≥(1/2+o(1))(n−ℓk−ℓ) for some ℓ≥k/2, then H contains a Hamilton (k−ℓ,ℓ)-cycle and this is asymptotically best possible.

Remarks on restricted fractional [formula omitted]-factors in graphs

Assume there exists a function h:E(G)→[0,1] such that g(x)≤∑e∈E(G),x∋eh(e)≤f(x) for every vertex x of G. The spanning subgraph of G induced by the set of edges {e∈E(G):h(e)>0} is called a fractional (g,f)-factor of G with indicator function h. Let M and N be two disjoint sets of independent edges of G satisfying |M|=m and |N|=n. We say that G possesses a fractional (g,f)-factor with the property E(m,n) if G contains a fractional (g,f)-factor with indicator function h such that h(e)=1 for each e∈M and h(e)=0 for each e∈N. In this article, we discuss stability number and minimum degree conditions for graphs to possess fractional (g,f)-factors with the property E(1,n). Furthermore, we explain that the stability number and minimum degree conditions declared in the main result are sharp.

Triangles in randomly perturbed graphs

AbstractWe study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$ -vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$ . We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$ , where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac.Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$ -universality.

Transversal Ck-factors in subgraphs of the balanced blow-up of Ck

AbstractFor a subgraph $G$ of the blow-up of a graph $F$ , we let $\delta ^*(G)$ be the smallest minimum degree over all of the bipartite subgraphs of $G$ induced by pairs of parts that correspond to edges of $F$ . Johansson proved that if $G$ is a spanning subgraph of the blow-up of $C_3$ with parts of size $n$ and $\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$ , then $G$ contains $n$ vertex disjoint triangles, and presented the following conjecture of Häggkvist. If $G$ is a spanning subgraph of the blow-up of $C_k$ with parts of size $n$ and $\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$ , then $G$ contains $n$ vertex disjoint copies of $C_k$ such that each $C_k$ intersects each of the $k$ parts exactly once. A similar conjecture was also made by Fischer and the case $k=3$ was proved for large $n$ by Magyar and Martin.In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of $G$ to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.