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Given a sequence $S=(s_1,s_2,\ldots,s_p)$, $p\geq 2$, of non-decreasing integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $p$ disjoint sets $V_1,\ldots, V_p$ such that any two distinct vertices of $V_i$…
A lambda colouring (or $L(2,1)-$colouring) of a graph is an assignment of non-negative integers (with minimum assignment $0$) to its vertices such that the adjacent vertices must receive integers at least two apart and vertices at distance…
A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study…
A Kneser representation KG(H) for a graph G is a bijective assignment of hyperedges of a hypergraph H to the vertices of G such that two vertices of G are adjacent if and only if the corresponding hyperedges are disjoint. In this paper, we…
Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…
Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph $H$ to be colored is locally embedded within the communication graph $G$. Besides generalizing…
Given a list assignment for a graph, list packing asks for the existence of multiple pairwise disjoint list colorings of the graph. Several papers have recently appeared that study the existence of such a packing of list colorings.…
A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…
The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…
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…
Consider edge colorings of digraphs where edges $v_1 v_2$ and $v_2 v_3$ have different colors. This coloring induces a vertex coloring by sets of edge colors, in which edge $v_1 v_2$ in the graph implies that the set color of $v_1$ contains…
A $k$-coloring of a graph $G$ is a $k$-partition $\Pi=\{S_1,\ldots,S_k\}$ of $V(G)$ into independent sets, called \emph{colors}. A $k$-coloring is called \emph{neighbor-locating} if for every pair of vertices $u,v$ belonging to the same…
For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots,…
The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph…
The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges…
An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits…
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…
The curling number of a graph G is defined as the number of times an element in the degree sequence of G appears the maximum. Graph colouring is an assignment of colours, labels or weights to the vertices or edges of a graph. A colouring…
Given a sequence \( S = (s_1, s_2, \ldots, s_k) \) of positive integers satisfying \( s_1 \leq s_2 \leq \dots \leq s_k \), an \( S \)-packing coloring of a graph \( G \) is a partition of \( V(G) \) into \( k \) subsets \( V_1, V_2, \dots,…
A signed graph is a pair $(G, \sigma)$, where $G$ is a graph and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend…