Related papers: Every subcubic graph is packing $(1,1,2,2,3)$-colo…

The chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which $G$ admits an edge $k$-coloring such that any two adjacent edges have distinct colors. The strong chromatic index $\chi'_s(G)$ of $G$ is the smallest $k$ such that…

Let $G$ be an edge-coloured graph. The minimum colour degree $ \delta^c(G) $ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly…

Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic…

An equitable coloring of a graph is a proper coloring where the sizes of any two distinct color classes differ by at most one. The celebrated Chen-Lih-Wu Conjecture (CLWC for short) states that every connected graph $G$ that is neither an…

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be…

Let $G=(V,E)$ be a graph of order $n$ with chromatic number $\chi(G)$. Let $ k \geq \chi(G) $ and $S \subseteq V$. Let $ C_0 $ be a $k$-coloring of the induced subgraph $ G[S] $. The coloring $C_0$ is called an extendable coloring, if $C_0$…

A mixed graph has a set of vertices, a set of undirected egdes, and a set of directed arcs. A proper coloring of a mixed graph $G$ is a function $c$ that assigns to each vertex in $G$ a positive integer such that, for each edge $uv$ in $G$,…

For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors.…

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of…

A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL…

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$.…

Inspired by the majority colorings and C-colorings, we introduce and study the majority C-coloring of graphs. In such a vertex coloring, every vertex shares its color with at least half of its neighbors. The maximum number of colors that…

A graph coloring has bounded clustering if each monochromatic component has bounded size. Equivalently, it is a partition of the vertices into induced subgraphs with bounded size components. This paper studies clustered colorings of graphs,…

We study two measures of uncolourability of cubic graphs, their colouring defect and perfect matching index. The colouring defect of a cubic graph $G$ is the smallest number of edges left uncovered by three perfect matchings; the perfect…

A graph $G$ is $(d_1,\ldots,d_k)$-colorable if its vertex set can be partitioned into $k$ sets $V_1,\ldots,V_k$, such that for each $i\in\{1, \ldots, k\}$, the subgraph of $G$ induced by $V_i$ has maximum degree at most $d_i$. The Four…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…

An incidence of a graph $G$ is a pair $(u,e)$ where $u$ is a vertex of $G$ and $e$ is an edge of $G$ incident with $u$. Two incidences $(u,e)$ and $(v,f)$ of $G$ are adjacent whenever (i) $u=v$, or (ii) $e=f$, or (iii) $uv=e$ or $uv=f$. An…

In this short note we show that every connected $2$-edge coloured cubic graph admits an $10$-colouring. This lowers the best known upper bound for the chromatic number of connected $2$-edge coloured cubic graphs.

For $p\in \mathbb{N}$, a coloring $\lambda$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $\lambda$ or there is a color that appears exactly…