相关论文: Packing coloring of graphs with long paths

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. In this short note we…

The star chromatic number on a graph is the minimum number of colors in a proper vertex coloring forbidding any $P_4$ with two colors (bicolored). This problem was introduced by Gr\"unbaum (1973) together with the acyclic coloring of…

A star coloring of a graph $G$ is a proper vertex coloring such that no path on four vertices is bicolored. The smallest integer $k$ for which $G$ admits a star coloring with $k$ colors is called the star chromatic number of $G$, denoted as…

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 $V_i$…

An injective colouring of a graph is a colouring in which every two vertices sharing a common neighbour receive a different colour. Chen, Hahn, Raspaud and Wang conjectured that every planar graph of maximum degree $\Delta \ge 3$ admits an…

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the…

Consider a graph whose vertices are colored in one of two colors, say black or white. A white vertex is called integrated if it has at least as many black neighbors as white neighbors, and similarly for a black vertex. The coloring as a…

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that vertices of $G$ can be partitioned into disjoint classes $X_1, ..., X_k$ where vertices in $X_i$ have pairwise distance greater than $i$. We…

Given a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, the mapping $f:V(G) \rightarrow \{1,\ldots,k\}$ is an $S$-packing $k$-coloring of $G$ if for any distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=i$…

The computation of short paths in graphs with arc lengths is a pillar of graph algorithmics and network science. In a more diverse world, however, not every short path is equally valuable. For the setting where each vertex is assigned to a…

A packing $k$-coloring is a natural variation on the standard notion of graph $k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, and any two vertices assigned a common color $c \in \{1, \ldots, k\}$ need to be at a…

The packing chromatic number $\chi$ $\rho$ (G) of a graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1 ,. .. , V k , in such a way that every two distinct vertices in V i are…

A Star Coloring of a graph G is a proper vertex coloring such that every path on four vertices uses at least three distinct colors. The minimum number of colors required for such a star coloring of G is called star chromatic number, denoted…

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…

For a nondecreasing sequence of integers $S=(s_1, s_2, \ldots)$ an $S$-packing $k$-coloring of a graph $G$ is a mapping from $V(G)$ to $\{1, 2,\ldots,k\}$ such that vertices with color $i$ have pairwise distance greater than $s_i$. By…

Although it has recently been proved that the packing chromatic number is unbounded on the class of subcubic graphs, there exists subclasses in which the packing chromatic number is finite (and small). These subclasses include subcubic…

An $i$-independent set is a set of vertices whose pairwise distance is at least $i+1$. A proper coloring (resp. a square coloring) of a graph is a partition of its vertices into independent (resp. $2$-independent) sets. A packing…

In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…

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…

A connected component of a vertex-coloured graph is said to be colourful if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph $G$ and an…