Related papers: Hypergraphs with Zero Chromatic Threshold

In this paper, we generalize the concepts related to rainbow coloring to hypergraphs. Specifically, an $(n,r,H)$-local coloring is defined as a collection of $n$ edge-colorings, $f_v: E(K^{(r)}_n) \rightarrow [k]$ for each vertex $v$ in the…

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ whenever $G$ is an $F$-free graph of maximum degree $\Delta$. The…

In 1973, Erd\H{o}s and Simonovits asked whether every $n$-vertex triangle-free graph with minimum degree greater than $1/3 \cdot n$ is 3-colourable. This question initiated the study of the chromatic profile of triangle-free graphs: for…

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the…

The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and…

A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. Gr\'osz, Methuku and Tompkins in 2020 showed that for any graph $F$, there is an integer $r_0=r_0(F)$, such that for any $r\ge r_0$, any $r$-uniform…

Erd\H{o}s and Simonovits asked the following question: For an integer $c\geq 2$ and a family of non-bipartite graphs $\mathcal{F}$, what is the infimum of $\alpha$ such that any $\mathcal{F}$-free $n$-vertex graph with $n$ large enough and…

An $r$-uniform hypergraph, or $r$-graph, has density $|E(G)|/|V(G)^{(r)}|$. We say $\alpha$ is a jump for $r$-graphs if there is some constant $\delta=\delta(\alpha)$ such that, for each $\varepsilon>0$ and $n\geq r$, any sufficiently large…

The anti-Ramsey number $\mathrm{ar}(n,F)$ of an $r$-graph $F$ is the minimum number of colors needed to color the complete $n$-vertex $r$-graph to ensure the existence of a rainbow copy of $F$. We establish a removal-type result for the…

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is…

Let $S=\{n_1,n_2,...,n_t\}$ be a finite set of positive integers with $\min(S)\geq 3$ and $t\geq 2$. For any positive integers $s_1,s_2,...,s_t$, we construct a family of 3-uniform bi-hypergraphs ${\cal H}$ with the feasible set $S$ and…

Given a fixed graph $H$ and a constant $c \in [0,1]$, we can ask what graphs $G$ with edge density $c$ asymptotically maximize the homomorphism density of $H$ in $G$. For all $H$ for which this problem has been solved, the maximum is always…

We show that the chromatic index of a hypergraph $\mathcal{H}$ satisfies Berge-F\"uredi conjectured bound $\mathrm{q}(\mathcal{H})\le \Delta([\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\mathrm{ar}(\mathcal{H})$ or on the…

A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show…

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study…

We study hypergraphs which are uniquely determined by their chromatic, independence and matching polynomials. B. Bollob\'as, L. Pebody and O. Riordan (2000) conjectured (BPR-conjecture) that almost all graphs are uniquely determined by…

Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be $r$ different colors. Suppose we randomly and uniformly color the edges of $K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by…

Let $G$ be a simple graph with maximum degree denoted as $\Delta(G)$. An overfull subgraph $H$ of $G$ is a subgraph satisfying the condition $|E(H)| > \Delta(G)\lfloor \frac{1}{2}|V(H)| \rfloor$. In 1986, Chetwynd and Hilton proposed the…

One of the most interesting new developments in hypergraph colourings in these last few years has been Voloshin's notion of colourings of mixed hypergraphs. In this paper we shall study a specific instance of Voloshin's idea: a…

A {\it mixed hypergraph} ${\cal H}=({\cal V},{\cal C},{\cal D})$ consists of the vertex set ${\cal V}$ and two families of subsets of $2^{{\cal V}}$: the family ${\cal C}$ of co-edges and the family ${\cal D}$ of edges. ${\cal H}$ is said…