Related papers: Local rainbow colorings for various graphs

A result of Gy\'arf\'as says that for every $3$-coloring of the edges of the complete graph $K_n$, there is a monochromatic component of order at least $\frac{n}{2}$, and this is best possible when $4$ divides $n$. Furthermore, for all…

Let $G=(V,E)$ be a multigraph of maximum degree $\Delta$. The edges of $G$ can be colored with at most $\frac{3}{2}\Delta$ colors by Shannon's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\Delta$…

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $\Delta$ has a coloring with $O(\Delta^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known…

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that…

An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors…

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors.…

In this paper, we investigate three extensions of Ramsey numbers to other combinatorial settings. We first consider ordered Ramsey numbers. Here, we ask for a monochromatic copy of a linearly ordered graph $G$ in every $2$-edge-coloring of…

A $k$-coloring of a graph is an assignment of integers between $1$ and $k$ to vertices in the graph such that the endpoints of each edge receive different numbers. We study a local variation of the coloring problem, which imposes further…

A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex…

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…

An edge colored graph $G$ is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are…

For graphs $G$ and $H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H $ if any proper edge-coloring of $G$ contains a rainbow copy of $H$, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last…

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kyn\v{c}l and Valtr showed that $h(n) = O(n^2/\log^{1/4} n)$, and in the other…

For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\psi:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrow\psi(i)\psi(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all…

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$…

An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in…

In this paper, we introduce the notion of $t$-tone edge coloring. A $t$-tone edge $k$-coloring of a graph $G$ assigns to each edge of $G$ a set of $t$ distinct colors from $\{1,\dots,k\}$ such that any two edges at distance $d$ share fewer…

The extremal problem of hypergraph colorings related to Erd\H{o}s--Hajnal property $B$-problem is considered. Let $k$ be a natural number. The problem is to find the value of $m_k(n)$ equal to the minimal number of edges in an $n$-uniform…

Strengthening Hadwiger's conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$-minor is properly $(t-1)$-colorable, this is known as the Odd Hadwiger's conjecture. We prove a relaxation of the above…