Super-pancyclic hypergraphs and bipartite graphs

Kostochka, Alexandr; Luo, Ruth; Zirlin, Dara

doi:10.1016/j.jctb.2020.06.007

Cited by 11 publications

(15 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To be precise, we partition the vertex set into two parts X and Y , where X consists of vertices of degree at least d, and then apply the above results on maximum cycles to find few number of disjoint paths in the bipartite subgraph G(X, Y ) to cover all vertices in X; finally, an application of a lemma of Erdős et al in [3] (see Lemma 3.6) will ensure the desired long path. We would like to point out that it seems to be an incredible coincidence that the bounds we need in this argument are exactly what the recent result of Kostochka, Luo and Zirlin [7] provided. The case when n ≥ 2d + 2 will be handled differently, which is essentially reduced to the base case.…”

Section: Introductionmentioning

confidence: 54%

“…The first one is due to Jackson [6]. The following theorem, conjectured by Jackson [6] and proved by Kostochka, Luo and Zirlin [7] recently, extends the above theorem of Jackson to the setting of 2-connected graphs. Our proof actually needs some intermediate statements from the proof of [7].…”

Section: Preliminaries and Some Results On Bipartite Graphsmentioning

confidence: 86%

“…Our proof actually needs some intermediate statements from the proof of [7]. Let us give some notation used in [7] first. Let G be bipartite with parts X and Y which is not a forest.…”

Section: Preliminaries and Some Results On Bipartite Graphsmentioning

confidence: 99%

“…For that it is crucial for us to manage the base case when d + 1 ≤ n ≤ 2d + 1. It turns out in the proof of the base case that we make use of results on maximum cycles in bipartite graphs due to Jackson [6] and Kostochka, Luo and Zirlin [7] (see Theorems 3.1 and 3.2, respectively). To be precise, we partition the vertex set into two parts X and Y , where X consists of vertices of degree at least d, and then apply the above results on maximum cycles to find few number of disjoint paths in the bipartite subgraph G(X, Y ) to cover all vertices in X; finally, an application of a lemma of Erdős et al in [3] (see Lemma 3.6) will ensure the desired long path.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Extremal problems of Erdős, Faudree, Schelp and Simonovits on paths and cycles

Li,

Ma,

Ning

2021

Preprint

View full text Add to dashboard Cite

For positive integers n > d ≥ k, let φ(n, d, k) denote the least integer φ such that every n-vertex graph with at least φ vertices of degree at least d contains a path on k + 1 vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function φ(n, d, k), and conjectured that for any positive integers n > d ≥ k, it holds that φ(n, d, k) ≤ ⌊ k−1 2 ⌋⌊ n d+1 ⌋ + ǫ, where ǫ = 1 if k is odd and ǫ = 2 otherwise. In this paper we determine the value of the function φ(n, d, k) exactly. This confirms the above conjecture of Erdős et al. for all positive integers k = 4 and in a corrected form for the case k = 4. Our proof utilizes, among others, a lemma of Erdős et al. [3], a theorem of Jackson [6], and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin [7], where the latter two results concern maximum cycles in bipartite graphs. Besides, we construct examples to provide answers to two closely related questions raised by Erdős et al.

show abstract

Section: Introductionmentioning

confidence: 54%

Section: Preliminaries and Some Results On Bipartite Graphsmentioning

confidence: 86%

“…Our proof actually needs some intermediate statements from the proof of [7]. Let us give some notation used in [7] first. Let G be bipartite with parts X and Y which is not a forest.…”

Section: Preliminaries and Some Results On Bipartite Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extremal problems of Erdős, Faudree, Schelp and Simonovits on paths and cycles

Li,

Ma,

Ning

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…survey [10]. A similar notion for hypergraphs and a strengthening of it were recently considered in [8].…”

Section: Super-pancyclic Hypergraphs and Super-cyclic Bigraphsmentioning

confidence: 99%

Conditions for a Bigraph to be Super-Cyclic

Kostochka

Lavrov²,

Luo

et al. 2021

Electron. J. Combin.

Self Cite

View full text Add to dashboard Cite

A hypergraph $\mathcal H$ is super-pancyclic if for each $A \subseteq V(\mathcal H)$ with $|A| \geqslant 3$, $\mathcal H$ contains a Berge cycle with base vertex set $A$. We present two natural necessary conditions for a hypergraph to be super-pancyclic, and show that in several classes of hypergraphs these necessary conditions are also sufficient. In particular, they are sufficient for every hypergraph $\mathcal H$ with $ \delta(\mathcal H)\geqslant \max\{|V(\mathcal H)|, \frac{|E(\mathcal H)|+10}{4}\}$. We also consider super-cyclic bipartite graphs: those are $(X,Y)$-bigraphs $G$ such that for each $A \subseteq X$ with $|A| \geqslant 3$, $G$ has a cycle $C_A$ such that $V(C_A)\cap X=A$. Such graphs are incidence graphs of super-pancyclic hypergraphs, and our proofs use the language of such graphs.

show abstract