Related papers: Realising every colour distribution sequence with …

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

We show that for any positive integer $r$ there exists an integer $k$ and a $k$-colouring of the edges of $K_{2^{k}+1}$ with no monochromatic odd cycle of length less than $r$. This makes progress on a problem of Erd\H{o}s and Graham and…

A $k$-proper edge-coloring of a graph G is called adjacent vertex-distinguishing if any two adjacent vertices are distinguished by the set of colors appearing in the edges incident to each vertex. The smallest value $k$ for which $G$ admits…

Krenn, Gu and Zeilinger initiated the study of PMValid edge-colourings because of its connection to a problem from quantum physics. A graph is defined to have a PMValid $k$-edge-colouring if it admits a $k$-edge-colouring (i.e. an edge…

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of…

There is a huge difference in techniques and runtimes of distributed algorithms for problems that can be solved by a sequential greedy algorithm and those that cannot. A prime example of this contrast appears in the edge coloring problem:…

The $\!{}\bmod k$ chromatic index of a graph $G$ is the minimum number of colors needed to color the edges of $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1\!\!\pmod k$. Recently, the…

The generalized Ramsey number $r(G, H, q)$ is the minimum number of colors needed to color the edges of $G$ such that every isomorphic copy of $H$ has at least $q$ colors. In this note, we improve the upper and lower bounds on $r(K_{n, n},…

Given a graph G, a colouring is an assignment of colours to the vertices of G so that no two adjacent vertices are coloured the same. If all colour classes have size at most t, then we call the colouring t-bounded, and the t-bounded…

A $(4, 5)$-coloring of $K_n$ is an edge-coloring of $K_n$ where every $4$-clique spans at least five colors. We show that there exist $(4, 5)$-colorings of $K_n$ using $\frac 56 n + o(n)$ colors. This settles a disagreement between…

Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound…

Let $K_n^c$ be an edge-coloured complete graph on $n$ vertices. Let $\Delta_{\rm mon}(K_n^c)$ denote the largest number of edges of the same colour incident with a vertex of $K_n^c$. A properly coloured cycle is a cycle such that no two…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…

There has been extensive studies on the following question: given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a `colorful' copy of $H$, i.e., a copy of $H$ containing at most one edge…

An edge coloring $c$ of a graph $G$ is a royal $k$-edge coloring of $G$ if the edges of $G$ are assigned nonempty subsets of the set $\{1, 2, \ldots, k\}$ in such a way that the vertex coloring obtained by assigning the union of the colors…

Given a sequence $\mathbf{k} := (k_1,\ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges…

Let $\chi'_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 \pmod k$. Scott [{\em Discrete Math. 175}, 1-3…

For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does…

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…

Given a graph $H$ and an integer $k\ge1$, the Gallai-Ramsey number $GR_k(H)$ is defined to be the minimum integer $n$ such that every $k$-edge coloring of the complete graph $K_n$ contains either a rainbow (all different colored) triangle…