Related papers: Explicit tough Ramsey graphs

Let $r,s,t\geq2$ be integers. For $r$-graphs $G$ and $F_1,\dots,F_s$, we write $G\to(F_1,\dots,F_s)$ if every $s$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$-th color for some $1\leq i\leq s$. Let…

The universal homogeneous triangle-free graph, constructed by Henson and denoted $\mathcal{H}_3$, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with…

A relational structure is \emph{strongly indivisible} if for every partition $M = X_0 \sqcup X_1$, the induced substructure on $X_0$ or $X_1$ is isomorphic to $\mathcal{M}$. Cameron (1997) showed that a graph is strongly indivisible if and…

The toughness $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a cut set of vertices in}~G\}$ for $G\ncong K_n.$ The concept of toughness initially proposed by Chv$\mathrm{\acute{a}}$tal in 1973, which serves as a simple way to measure…

In Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red.…

We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices…

For a graph $G$, let $\mu_k(G):=\min~\{\max_{x\in S}d_G(x):~S\in \mathcal{S}_k\}$, where $\mathcal{S}_k$ is the set consisting of all independent sets $\{u_1,\ldots,u_k\}$ of $G$ such that some vertex, say $u_i$ ($1\leq i\leq k$), is at…

In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been…

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial…

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting…

A graph $G$ is semilinear of complexity $t$ if the vertices of $G$ are elements of $\mathbb{R}^{d}$ for some $d\in\mathbb{Z}^{+}$, and the edges of $G$ are defined by the sign patterns of $t$ linear functions…

Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic…

In the $m$-dimensional affine space $AG(m,q)$ over the finite field $\mathbb{F}_q$ of odd order $q$, the analogous of the Euclidean distance gives rise to a graph $\mathfrak{G}_{m,q}$ where vertices are the points of $AG(m,q)$ and two…

Let $G$ be a graph. We say that $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. We use $P_t$ and $C_t$ to denote a path…

We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of…

Let $k$ be a positive integer. A graph is said to be $(P_2 \cup kP_1)$-free if it does not contain $P_2 \cup kP_1$ as an induced subgraph. Recently, Ota and the author asked whether every 1-tough and $k$-connected $(P_2 \cup kP_1)$-free…

A graph $G$ is called a $2K_2$-free graph if it does not contain $2K_2$ as an induced subgraph. In 2014, Broersma, Patel and Pyatkin showed that every 25-tough $2K_2$-free graph on at least three vertices is Hamiltonian. Recently, Shan…

Katona and Varga showed that for any rational number $t \in (1/2,1]$, no chordal graph is minimally $t$-tough, while Katona and Khan characterized all minimally $t$-tough, chordal graphs with $t \le 1/2$. We conjecture that no chordal graph…