Related papers: Equitable Dominator Coloring of Graphs

We introduce and study Fair and Tolerant colorings (FAT colorings), where each vertex tolerates a given fraction of same-colored neighbors while fairness is preserved across the other coloring classes. Moreover, we define the FAT chromatic…

Several authors modelled networks ad hoc by oriented or disoriented graphs, whereby the problem of allowance (allocation) of the frequencies at the level of the network was transformed into coloring problem of nodes in the graph. Graph…

This paper investigates the semi-streaming complexity of \textit{$k$-partial coloring}, a generalization of proper graph coloring. For $k \geq 1$, a $k$-partial coloring requires that each vertex $v$ in an $n$-node graph is assigned a color…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

With respect to a proper colouring of a graph $G$, we know that $\delta(G) \leq \chi(G) \leq \Delta(G)+1$. If distinct colours represent distinct technology types to be located at vertices the question arises on how to place at least one of…

A graceful k-coloring of a non-empty graph $G=(V,E)$ is a proper vertex coloring $f:V(G)\rightarrow\lbrace 1,2,...,k \rbrace$, $k\geq 2$, which induces a proper edge coloring $f^{*}:E(G)\rightarrow\lbrace 1, 2, . . . , k-1 \rbrace $ defined…

A vertex coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, shown by $D(G)$, is the smallest number of colors required for such a…

Irredundance coloring of $G$ is a proper coloring in which there exists a maximal irredundant set $R$ such that all the vertices of $R$ have different colors. The minimum number of colors required for an irredundance coloring of $G$ is…

A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the…

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…

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so…

DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo\v{r}\'{a}k and Postle. Several known bounds for the list chromatic number of a graph $G$, $\chi_\ell(G)$, also hold for the…

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating…

$\DeclareMathOperator{\chicen}{\chi_{\mathrm{cen}}}\DeclareMathOperator{\chilin}{\chi_{\mathrm{lin}}}$ 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 set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a…

A (not necessarily proper) vertex coloring of a graph $G$ with color classes $V_1$, $V_2$, $\dots$, $V_k$, is said to be a {\it Fair And Tolerant vertex coloring of $G$ with $k$ colors}, whenever $V_1$, $V_2$, $\dots$, $V_k$ are nonempty…

In this paper, a $k$-edge-coloring of $G$ is any mapping $c:E(G)\longrightarrow [k]$. The edge-coloring $c$ of $G$ naturally defines a vertex-coloring $\sigma_{c}: V(G) \to \mathbb{N}$, where $\sigma_{c}(v)=\sum_{u\in N_G(v)}c(vu)$ for…

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of $k > 0$ colors. Each different ruleset specifies that game's coloring constraints. This paper investigates six impartial…

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with…

We introduce the notion of a properly ordered coloring (POC) of a weighted graph, that generalizes the notion of vertex coloring of a graph. Under a POC, if $xy$ is an edge, then the larger weighted vertex receives a larger color; in the…