Related papers: Total Difference Chromatic Numbers of Graphs
A graph G is distinguished if its vertices are labelled by a map \phi: V(G) \longrightarrow {1,2,...,k} so that no graph automorphism preserves \phi. The distinguishing number of G is the minimum number k necessary for \phi to distinguish…
The minimum number of total independent partition sets of $V \cup E$ of a graph $G=(V,E)$ is called the \emph{total chromatic number} of $G$, denoted by $\chi''(G)$. If the difference between cardinalities of any two total independent sets…
For a graph $G$, the $k$-colouring graph of $G$ has vertices corresponding to proper $k$-colourings of $G$ and edges between colourings that differ at a single vertex. The graph supports the Glauber dynamics Markov chain for $k$-colourings,…
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…
By a graph we mean a finite undirected graph having multiple edges but no loops. Given a graph property $\mathcal{P}$, a $\mathcal{P}$-coloring of a graph $G$ with color set $C$ is a mapping $\f:V(G)\to C$ such that for each color $c\in C$…
A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold $\theta(G)$ of a graph $G$ is the minimum number of colors $k$ required that any arbitrary $k$-coloring…
Let $k \ge 1$ be an integer and let $G$ be a nonempty simple graph. An \emph{edge-$k$-coloring} $\varphi$ of $G$ is an assignment of colors from $\{1,\ldots,k\}$ to the edges of $G$ such that no two adjacent edges receive the same color.…
A total prime labeling of a graph of order $n$ is an extension of a prime labeling in which we distinctly label the vertices and edges. The goal of the labeling is for adjacent vertex labels to be relatively prime, and for each vertex of…
Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$.…
A $k$-coloring of a graph $G$ is a partition of the set of vertices of $G$ into $k$ independent sets, which are called colors. A $k$-coloring is neighbor-locating if any two vertices belonging to the same color can be distinguished from…
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…
A vertex coloring of a given graph $G$ is conflict-free if the closed neighborhood of every vertex contains a unique color (i.e. a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the…
By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]=\{1,2,\ldots, n\}$ for some integers $n\ge k\ge 1$, and in which two such sets are adjacent if and only if they realise a certain order type…
The $k$th power $G^k$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^k$ if the distance between $u$ and $v$ in $G$ is at most $k$. Let $\chi(H)$ and $\chi_l(H)$ be the chromatic number…
Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…
A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number…
A $k$-dispersed labelling of a graph $G$ on $n$ vertices is a labelling of the vertices of $G$ by the integers $1, \dots , n$ such that $d(i,i+1) \geq k$ for $1 \leq i \leq n-1$. $DL(G)$ denotes the maximum value of $k$ such that $G$ has a…
Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing…
A proper 2-tone $k$-coloring of a graph is a labeling of the vertices with elements from $\binom{[k]}{2}$ such that adjacent vertices receive disjoint labels and vertices distance 2 apart receive distinct labels. The 2-tone chromatic number…
A graph $G$ is called chromatic-choosable if $\chi(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that…