Related papers: Circular chromatic number of signed graphs
A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…
An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by…
A star edge-coloring of a graph $G$ is a proper edge-coloring without bichromatic paths or cycles of length four. The smallest integer $k$ such that $G$ admits a star edge-coloring with $k$ colors is the star chromatic index of $G$. In the…
The harmonious chromatic number of a graph $G$ is the minimum number of colors that can be assigned to the vertices of $G$ in a proper way such that any two distinct edges have different color pairs. This paper gives various results on…
A \emph{star coloring} of a graph $G$ is a proper vertex-coloring such that no path on four vertices is $2$-colored. The minimum number of colors required to obtain a star coloring of a graph $G$ is called star chromatic number and it is…
For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…
The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic…
A strong edge colouring of a graph is an assignment of colours to the edges of the graph such that for every colour, the set of edges that are given that colour form an induced matching in the graph. The strong chromatic index of a graph…
A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index…
A complete $k$-coloring of a graph $G$ is a (not necessarily proper) $k$-coloring of the vertices of $G$, such that each pair of different colors appears in an edge. A complete $k$-coloring is also called connected, if each color class…
The chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum $k$ such that $G$ admits a $k$-coloring of its vertex set in such a way that each color class is an independent set (a set of pairwise non-adjacent vertices). The…
Let $G$ be a connected graph with maximum degree $\Delta \ge 3$. We investigate the upper bound for the chromatic number $\chi_\gamma(G)$ of the power graph $G^\gamma$. It was proved that $\chi_\gamma(G)…
Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined…
An ordered graph $G$ is a graph whose vertex set is a subset of integers. The edges are interpreted as tuples $(u,v)$ with $u < v$. For a positive integer $s$, a matrix $M \in \mathbb{Z}^{s \times 4}$, and a vector $\mathbf{p} =…
The strong chromatic number $\chi_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of…
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…
Let $\mathcal{C} = \{c_1,c_2, c_3, \ldots,c_k\}$ be a certain type of proper $k$-colouring of a given graph $G$ and $\theta(c_i)$ denote the number of times a particular colour $c_i$ is assigned to the vertices of $G$. Then, the colouring…
A signed tree-coloring of a signed graph $(G,\sigma)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,\sigma)$ whose vertex set is the set of…
The $k$-th exact-distance graph, of a graph $G$ has $V(G)$ as its vertex set, and $xy$ as an edge if and only if the distance between $x$ and $y$ is (exactly) $k$ in $G$. We consider two possible extensions of this notion for signed graphs.…
In 2020, Behr defined the problem of edge coloring of signed graphs and showed that every signed graph $(G, \sigma)$ can be colored using exactly $\Delta(G)$ or $\Delta(G) + 1$ colors, where $\Delta(G)$ is the maximum degree in graph $G$.…