Related papers: On the difference between the chromatic and cochro…

In this article we consider a problem related to two famous combinatorial topics. One of them concerns the chromatic number of the space. The other deals with graphs having big girth (the length of the shortest cycle) and large chromatic…

A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$…

\noindent The b-chromatic number of a graph $G$, denoted by $\phi(G)$, is the largest integer $k$ that $G$ admits a proper coloring by $k$ colors, such that each color class has a vertex that is adjacent to at least one vertex in each of…

The star chromatic index of a multigraph $G$, denoted $\chi'_{st}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bicolored. We survey the results of determining…

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic…

Let $c$ be a proper $k$-coloring of a connected graph $G$ and $\Pi=(C_1,C_2,...,C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\Pi$ is defined to be…

If we fix a spanning subgraph $H$ of a graph $G$, we can define a chromatic number of $H$ with respect to $G$ and we show that it coincides with the chromatic number of a double covering of $G$ with co-support $H$. We also find a few…

Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $\chi_d^t(G)$…

A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using…

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been…

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the…

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

Let G be a simple connected graph. Then chi L(G) and chi D(G) will denote the locating chromatic number and the distinguishing chromatic number of G, respectively. In this paper, we investigate a comparison between chi L(G) and chi D(G). In…

Let $G$ be a graph with maximum degree $\Delta(G)$ and maximum multiplicity $\mu(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $\Delta(G)+\mu(G)$. The distance between two edges $e$ and…

For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping $f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each edge $uv$ of $G$. The smallest integer $k$ for which there is a proper $k$-coloring…

Here we prove that if G has independence no. 2 and clique size omega with omega less than or equal to 11, then (1) chromatic no. is less than or equal to (omega2+12omega-13)/8, if omega is odd, and (2) chromatic no. is less than or equal to…

Chromatic-choosablility is a notion of fundamental importance in list coloring. A graph is chromatic-choosable when its chromatic number is equal to its list chromatic number. In 1990, Kostochka and Sidorenko introduced the list color…

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 distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. We say that a graph $G$ is $d$-distinguishing critical, if…