Related papers: Modified vertex Folkman numbers

Let $G$ be a graph and $a_1, ..., a_s$ be positive integers. Then $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i \in \{1, ..., s\}$, such that there is a…

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for any $s$-coloring of the vertices of $G$ there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex…

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$.…

For a graph $G$ and integers $a_i\ge 1$, the expression $G \rightarrow (a_1,\dots,a_r)^v$ means that for any $r$-coloring of the vertices of $G$ there exists a monochromatic $a_i$-clique in $G$ for some color $i \in \{1,\cdots,r\}$. The…

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every $s$-coloring of the vertices of $G$ there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex…

For a graph $G$ the symbol $G\tov(a_1,...,a_r)$ means that in every $r$-coloring of the vertices of $G$ for some $i\in\{1,...,r\}$ there exists a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers \[…

For graph $G$ and integers $a_1 \ge \cdots \ge a_r \ge 2$, we write $G \rightarrow (a_1 ,\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \in \{1,…

We present some new constructive upper bounds based on product graphs for generalized vertex Folkman numbers. They lead to new upper bounds for some special cases of generalized edge Folkman numbers, including $F_e(K_3,K_4-e; K_5) \leq 27$…

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority…

For an undirected simple graph $G$, we write $G \rightarrow (H_1, H_2)^v$ if and only if for every red-blue coloring of its vertices there exists a red $H_1$ or a blue $H_2$. The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is…

For a graph $G$ and integers $a_i \geq 1$, we say that $G \xrightarrow[]{} (a_1, \ldots, a_k)^v$ if in any $k$-coloring of $G$'s vertices there exists a monochromatic $a_i$-clique for some color $i \in \{1,\ldots,k\}$. $G \xrightarrow[]{}…

Given positive integers $k \leq m$ and a graph $G$, a family of lists $L = \{L(v) : v \in V(G)\}$ is said to be a random $(k,m)$-list-assignment if for every $v \in V(G)$ the list $L(v)$ is a subset of $\{1, \ldots, m\}$ of size $k$, chosen…

A (not necessarily proper) vertex coloring of a graph $G$ with color classes $V_1$, $V_2$, $\dots$, $V_k$, is said to be a {\it Fair And Tolerant vertex coloring of $G$ with $k$ colors}, whenever $V_1$, $V_2$, $\dots$, $V_k$ are nonempty…

A coloring of a graph $G=(V,E)$ is a partition $\{V_1, V_2, \ldots, V_k\}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every…

The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and $E(G)$ is…

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the…

Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of…

The vertex Folkman number $F_v(s,t;k)$ is the smallest $n$ for which there exists a $K_k$-free graph on $n$ vertices whose vertices cannot be $2$-colored without producing a monochromatic copy of $K_s$ or $K_t$. We show $F_v(3,3;5)=8$. The…

Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is $m$-\textit{coloured} if each of the $m$ colours is used. For an $m$-colouring $\Delta$ of $\mathbb{N}^{(2)}$, the complete graph on $\mathbb{N}$, we…