On r-uniform hypergraphs with circumference less than r

Abstract

We show that for each k ≥ 4 and n > r ≥ k + 1, every n-vertex r-uniform hypergraph with no Berge cycle of length at least k has at most (k−1)(n−1) r edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Győri, Katona and Lemons that for n > r ≥ k ≥ 3, every n-vertex r-uniform hypergraph with no Berge path of length k has at most (k−1)n r+1 edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least 2k, and then translate the results into the language of multi-hypergraphs.Mathematics Subject Classification: 05C35, 05C38.Later, the remaining case k = r + 1 was resolved by Davoodi, Győri, Methuku, and Tompkins.Theorem 1.5 (Davoodi, Győri, Methuku and Tompkins [1]). Let k = r + 1 > 2, and let H be an n-vertex r-graph with no Berge-path of length k. Then e(H) ≤ n.Furthermore, the bounds in these three theorems are sharp for each k and r for infinitely many n.Very recently, several interesting results were obtained for Berge paths and cycles for k ≥ r + 1. First, Győri, Methuku, Salia, Tompkins, and Vizer [9] proved an asymptotic version of the Erdős-Gallai theorem for Berge-paths in connected hypergraphs whenever r is fixed and n and k tend to infinity. For Berge-cycles, the exact result for k ≥ r + 3 was obtained in [5]: Theorem 1.6 (Füredi, Kostochka and Luo [5]). Let k ≥ r + 3 ≥ 6, and let H be an n-vertex r-graph with no Berge-cycles of length k or longer. Then e(H) ≤ n−1 k−2 k−1 r . This theorem is a hypergraph version of Theorem 1.2 and an analog of Theorem 1.4. It also somewhat refines Theorem 1.4 for k ≥ r + 3. Later, Ergemlidze, Győri, Methuku, Salia, Thompkins, and Zamora [3] extended the results to to k ∈ {r + 1, r + 2}: Theorem 1.7 (Ergemlidze et al. [3]). If k ≥ 4 and H is an n-vertex r-graph with no Berge-cycles of length k or longer, then k = r + 1 and e(H) ≤ n − 1, or k = r + 2 and e(H) ≤ n−1 k−2 k−1 r .The goal of this paper is to prove a hypergraph version of Theorem 1.2 for r-graphs with no Berge-cycles of length k or longer in the case k ≤ r. Our result is an analog of Theorem 1.3 and yields a refinement of it.Our approach is to consider bipartite graphs in which the vertices in one of the parts have degrees at least r, and to analyze the structure of such graphs with circumference less than 2k. After that, we apply the obtained results to the incidence graphs of r-uniform hypergraphs. In this way, our methods differ from those of [8],[1], [9], and [5]. Notation and results Hypergraph notationThe lower rank of a multi-hypergraph H is the size of a smallest edge of H.In view of the structure of our proof, it is more convenient to consider hypergraphs with lower rank at least r instead of r-uniform hypergraphs. It also yields formally stronger statements of the results.The incidence graph G(H) of a multi-hypergraph H = (V, E) is the bipartite graph with parts V and E where v ∈ V is adjacent to e ∈ E iff in H vertex v belongs to edge e.There are several versions of connectivity of hypergraphs. We will call a m...

On 2-Connected Hypergraphs with No Long Cycles

“…Theorem 1.13 (Kostochka and Luo [15]). Let k ≥ 4, r ≥ k + 1, and let H be an n-vertex 2connected, r-uniform hypergraph with no Berge cycle of length k or longer.…”

“…This theorem is a hypergraph version of Theorem 1.4 for k ≥ r + 3. The case of k ≤ r − 1 was resolved by Kostochka and Luo [15].…”

“…For r ≥ k + 1, bounds for 2-connected hypergraphs stronger than for the general case were found in [15], although they are not known to be sharp.…”

“…Denote H := ∂ 2 H and let Q be the t-core of H (that is, the resulting graph from applying tdisintegration to H ). If H is t-disintegrable, i.e., Q is empty, then N Sp (H , r) ≤ f (|V (H )|, k, r, t) and so by (15), we get |E(H)| ≤ f (n, k, r, t). So we may assume that Q is non-empty.…”

“…3. Note that here we use r − -graphs to prove a bound for r-graphs when k > r and in [15] we used r + -graphs (i.e. hypergraphs with the lower rank at least r) in the case k < r.…”