Codegree Thresholds for Covering 3-Uniform Hypergraphs

Falgas–Ravry, Victor; Zhao, Yi

doi:10.1137/15m1051452

Cited by 10 publications

(17 citation statements)

References 29 publications

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“…(see [14]), where χ(F) is the chromatic number of F. Han, Zang and Zhao [14] studied the vertexdegree variant of the covering problem, for complete (3, 3)-graphs K. Falgas-Ravry and Zhao [11] studied c(n, F) when F is K 3 4 , K 3 4 with one edge removed, K 3 5 with one edge removed and other 3-graphs.…”

Section: Covering Thresholdsmentioning

confidence: 99%

Covering and tiling hypergraphs with tight cycles

Han

Sanhueza‐Matamala

2020

Combinator. Probab. Comp.

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A k-uniform tight cycle $C_s^k$ is a hypergraph on s > k vertices with a cyclic ordering such that every k consecutive vertices under this ordering form an edge. The pair (k, s) is admissible if gcd (k, s) = 1 or k / gcd (k,s) is even. We prove that if $s \ge 2{k^2}$ and H is a k-uniform hypergraph with minimum codegree at least (1/2 + o(1))|V(H)|, then every vertex is covered by a copy of $C_s^k$ . The bound is asymptotically sharp if (k, s) is admissible. Our main tool allows us to arbitrarily rearrange the order in which a tight path wraps around a complete k-partite k-uniform hypergraph, which may be of independent interest. For hypergraphs F and H, a perfect F-tiling in H is a spanning collection of vertex-disjoint copies of F. For $k \ge 3$ , there are currently only a handful of known F-tiling results when F is k-uniform but not k-partite. If s ≢ 0 mod k, then $C_s^k$ is not k-partite. Here we prove an F-tiling result for a family of non-k-partite k-uniform hypergraphs F. Namely, for $s \ge 5{k^2}$ , every k-uniform hypergraph H with minimum codegree at least (1/2 + 1/(2s) + o(1))|V(H)| has a perfect $C_s^k$ -tiling. Moreover, the bound is asymptotically sharp if k is even and (k, s) is admissible.

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Section: Covering Thresholdsmentioning

confidence: 99%

Covering and tiling hypergraphs with tight cycles

Han

Sanhueza‐Matamala

2020

Combinator. Probab. Comp.

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“…A tight r-uniform t-cycle C (r) t is an r-graph with a cyclic ordering of its t vertices such that every r consecutive vertices form an edge under this ordering. Falgas-Ravry and Zhao [21] determined c 2 (F), where F is K (3) 4 ,…”

Section: Introductionmentioning

confidence: 99%

“…This stands in contrast to the situation for ordinary graphs, where existence and covering thresholds essentially coincide. Given an r-graph F, Falgas-Ravry and Zhao [21] introduced the notion of an F-covering, which is intermediate between that of the existence of a single copy of F and the existence of an F-tiling.…”

mentioning

confidence: 99%

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

Falgas–Ravry

Markström

Zhao

2020

Combinator. Probab. Comp.

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We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

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“…In this note, we focus on the problem to determine the exact value of c 2 (n, K − t ) when t = 4 and 5. Falgas-Ravry and Zhao [2] determined the exact value of c 2 (n, K 4 ) for n > 98 and gave lower and upper bounds of c 2 (n, K − 4 ) and c 2 (n, K − 5 ). More specifically, they proved the following theorem.…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, they proved the following theorem. Theorem 1.1 (Theorem 1.2 in [2]). Suppose n = 6m + r for some r ∈ {0, 1, 2, 3, 4, 5} and m ∈ N with n 7.…”

Section: Introductionmentioning

confidence: 99%

Exact Minimum Codegree Thresholds for $K_4^-$-Covering and $K_5^-$-Covering

Yu¹,

Hou

Ma³

et al. 2020

Electron. J. Combin.

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Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let $c_2(n,F)$ be the minimum integer $t$ such that every 3-graph with minimum codegree greater than $t$ has an $F$-covering. In this note, we answer an open problem of Falgas-Ravry and Zhao (SIAM J. Discrete Math., 2016) by determining the exact value of $c_2(n, K_4^-)$ and $c_2(n, K_5^-)$, where $K_t^-$ is the complete $3$-graph on $t$ vertices with one edge removed.

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Codegree Thresholds for Covering 3-Uniform Hypergraphs

Abstract: Given two 3-uniform hypergraphs F and G, we say that G has an F -covering if we can cover

Cited by 10 publications

References 29 publications

Covering and tiling hypergraphs with tight cycles

Covering and tiling hypergraphs with tight cycles

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

Exact Minimum Codegree Thresholds for $K_4^-$-Covering and $K_5^-$-Covering

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