相关论文: Packing coloring of graphs with long paths
We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…
We study quantum analogs of graph colorings and chromatic number. Initially defined via an interactive protocol, quantum colorings can also be viewed as a natural operator relaxation of graph coloring. Since there is no known algorithm for…
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…
A \textit{$k$-total coloring} of a graph $G$ is an assignment of $k$ colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer $k$ for which…
{\emph A star edge-coloring} of a graph is a proper edge-coloring without bichromatic paths and cycles of length four. In this paper, we consider the list version of this coloring and prove that the list star chromatic index of every…
For a non-decreasing sequence of positive integers $S = (s_1,s_2,\ldots)$, the {\em $S$-packing chromatic number} $\chi_S(G)$ of $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $X_i$, $i \in…
A vertex coloring of a graph G is called a 2-distance coloring if any two vertices at a distance at most 2 from each other receive different colors. Suppose that G is a planar graph with a maximum degree at most 5. We prove that G admits a…
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph…
$\DeclareMathOperator{\chicen}{\chi_{\mathrm{cen}}}\DeclareMathOperator{\chilin}{\chi_{\mathrm{lin}}}$ A centred colouring of a graph is a vertex colouring in which every connected subgraph contains a vertex whose colour is unique and a…
A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} (MC-coloring, for short) if there is a…
The packing chromatic number $\chi_\rho(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set $V(G)$ can be partitioned into disjoint classes $X_1, ..., X_k$, where vertices in $X_i$ have pairwise distance greater than…
It was conjectured by Steinberg in 1976 that planar graphs without cycles of length 4 or 5 are 3-colorable. This conjecture attracted a substantial amount of attention and was finally refuted by Cohen-Addad, Hebdige, Kr\'{a}l', Li and…
Vertex coloring of a graph $G$ with $n$-colors can be equivalently thought to be a graph homomorphism (edge preserving vertex mapping) of $G$ to the complete graph $K_n$ of order $n$. So, in that sense, the chromatic number $\chi(G)$ of $G$…
In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two…
An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge…
A model named `Colored Percolation' has been introduced with its infinite number of versions in two dimensions. The sites of a regular lattice are randomly occupied with probability $p$ and are then colored by one of the $n$ distinct colors…
A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al.,…
Let $G$ be a graph with a set of precolored vertices, and let us be given an integer distance parameter $d$ and a set of integer demands $d_1,\dots,d_c$. The Distance Precoloring Extension with Demands (DPED) problem is to compute a vertex…
The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same. We show…