Related papers: A note on $\bar{X}$-coloring and $\hat{A}$-colorin…

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…

A hypergraph $\mathcal{H}$ on $n$ vertices and $m$ edges is said to be {\it nearly-intersecting} if every edge of $\mathcal{H}$ intersects all but at most polylogarthmically many (in $m$ and $n$) other edges. Given lists of colors…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We show that every planar graph without cycles of length 4 or 5 is…

The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $\phi$ with $\phi(v)\in L(v)$ for every vertex $v$ of $G$,…

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…

For planar graphs, we consider the problems of \emph{list edge coloring} and \emph{list total coloring}. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total…

It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For…

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

A {\it list assignment} $L$ of a graph $G$ is a function that assigns a set (list) $L(v)$ of colors to every vertex $v$ of $G$. Graph $G$ is called {\it $L$-list colorable} if it admits a vertex coloring $\phi$ such that $\phi(v)\in L(v)$…

Let $G$ be a graph and $\mathcal {S}$ be a subset of $Z$. A vertex-coloring $\mathcal {S}$-edge-weighting of $G$ is an assignment of weight $s$ by the elements of $\mathcal {S}$ to each edge of $G$ so that adjacent vertices have different…

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall…

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a…

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find…

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $\pi$ that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most…

Coloring the vertices of a graph G subject to given conditions can be considered as a random experiment and corresponding to this experiment, a discrete random variable X can be defined as the colour of a vertex chosen at random, with…

Given two graphs $F$ and $G$, an $F$-WORM coloring of $G$ is an assignment of colors to its vertices in such a way that no $F$-subgraph of $G$ is monochromatic or rainbow. If $G$ has at least one such coloring, then it is called $F$-WORM…

An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal…

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$-admissible if admits a spanning tree in which the distance between any two adjacent vertices of $G$…