Related papers: On a new problem about the local irregularity of g…

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some…

Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if…

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $\varphi$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$…

It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tilde{\Theta}\left(n ^ {\frac{1}{r+1}} \right)$ (that…

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the…

An edge coloring of a graph $G$ is called conflict-free if, for every edge, its closed neighborhood contains a color that appears exactly once. The least number of colors required for such a coloring is the conflict-free chromatic index of…

The 1-2-3 Conjecture, introduced by Karo\'nski, {\L}uczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph $G$ different from $K_2$, we can turn $G$ into a locally irregular multigraph $M(G)$,…

The star chromatic index of a multigraph $G$, denoted $\chi'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining…

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$.…

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $\chi'_a(G)$, is the smallest number of colors allowing…

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

A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For 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 $G$ be a graph and $R\subseteq V(G)$. A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an $R$-sequential $t$-coloring if the edges incident to each vertex $v\in R$ are colored by the colors $1,\ldots,d_{G}(v)$,…

Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident…

Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be…