Related papers: Improved bounds for centered colorings
For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much…
A {\it proper conflict-free $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing exactly once on its neighborhood. This notion was formally introduced by Fabrici et al., who proved that…
A lower bound is obtained for the greatest possible number of colors in an interval colourings of some regular graphs.
We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if $G$ is a connected graph with maximum degree $\Delta(G) \geq 4$ that is not a complete…
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…
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We…
A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest…
In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while…
A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $\varphi:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one…
In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$…
We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to $t$-core (subgraph induced by the vertices $v$ with $d(v)+\mu(v)> \Delta+t$), and find a sufficient condition for $(\Delta+t)$-edge-coloring.…
We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that…
A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…
The Grundy and the {\rm b}-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph $G$, denoted by $\Gamma(G)$ is the worst case behavior of greedy (First-Fit) coloring procedure for $G$ and the {\rm…
The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering $\sigma= v_1,\dots,v_n$, the "First-Fit" greedy colouring algorithm colours…
For any graph $G$, the Grundy (or First-Fit) chromatic number of $G$, denoted by $\Gamma(G)$ (also $\chi_{_{\sf FF}}(G)$), is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$.…
We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel…
A total coloring of a graph $G$ is a coloring of the vertices and edges such that two adjacent or incident elements receive different colors. The minimum number of colors required for a total coloring of a graph $G$ is called the total…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…