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Register a new userConnected Hypergraphs without long Berge paths
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We generalize a result of Balister, Gy{H{o}}ri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an $n$-vertex, $r$-uniform, connected, hypergraph with the maximum number of hyperedges, without a $k$-Berge-path, where $n geq N_{k,r}$, $kgeq 2r+13>17$.
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In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph $mathcal{H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of $mathcal{H}$. $mathcal{H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Turan number of Berge hypergraphs to the $t$-wise Berge hypergraphs case. We asymptotically determine the Turan number of $t$-heavy and $t$-wise Berge copies of long paths and cycles and exactly determine the Turan number of $t$-heavy and $t$-wise Berge copies of cliques. In the case of 3-uniform hypergraphs, we consider the problem in more details and obtain additional results.
For a fixed set of positive integers $R$, we say $mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $mathcal{H}$ is emph{covering} if every vertex pair of $mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $mathcal{H}$ is called a textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) to E(mathcal{H})$ such that for every $e in E(G)$, $e subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the emph{cover Ramsey number}, denoted as $hat{R}^R(BG_1, BG_2)$, as the smallest integer $n_0$ such that for every covering $R$-uniform hypergraph $mathcal{H}$ on $n geq n_0$ vertices and every $2$-edge-coloring (blue and red) of $mathcal{H}$ , there is either a blue Berge-$G_1$ or a red Berge-$G_2$ subhypergraph. We show that for every $kgeq 2$, there exists some $c_k$ such that for any finite graphs $G_1$ and $G_2$, $R(G_1, G_2) leq hat{R}^{[k]}(BG_1, BG_2) leq c_k cdot R(G_1, G_2)^3$. Moreover, we show that for each positive integer $d$ and $k$, there exists a constant $c = c(d,k)$ such that if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $hat{R}^{[k]}(BG,BG) leq cn$.
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $max{5,2lceil{frac{t-1}{2}}rceil-2}$ many colors. If the input graph is triangle-free, we only need $max{4,lceil{frac{t-1}{2}}rceil+1}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.
In this paper, we consider maximum possible value for the sum of cardinalities of hyperedges of a hypergraph without a Berge $4$-cycle. We significantly improve the previous upper bound provided by Gerbner and Palmer. Furthermore, we provide a construction that slightly improves the previous lower bound.
Given integers $k,j$ with $1le j le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the `Pathfinder algorithm, a depth-first search algorithm which discovers $j$-tight paths in a $k$-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long $j$-tight path.
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