Related papers: On Cartesian products of signed graphs
Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization…
This paper studies the circular coloring of signed graphs. A signed graph is a graph with a signature that assigns a sign to each edge, either positive or negative. This paper studies circular coloring and a circular chromatic number of…
There are many concepts of signed graph coloring which are defined by assigning colors to the vertices of the graphs. These concepts usually differ in the number of self-inverse colors used. We introduce a unifying concept for this kind of…
In this work, we study the $\delta$-chromatic number of a graph which is the chromatic number of the $\delta$-complement of a graph. We give a structure of the $\delta$-complements and sharp bounds on the $\delta$-chromatic numbers of the…
A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of (more than one) strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the…
A signed graph $(G, \sigma)$ is a graph $G$ along with a function $\sigma: E(G) \to \{+,-\}$. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A…
Let $\Gamma=(V,E)$ be a finite simple graph. A matching $M \subseteq E$ is positive if there exists a weight function on $V$ such that the matching $M$ is characterized by those edges with positive weights. A positive matching decomposition…
We define the Cartesian, Categorical, and Lexicographic, and Strong products of quantum graphs. We provide bounds on the quantum chromatic number of these products in terms of the quantum chromatic number of the factors. To adequately…
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…
Gravier et al. investigated the identifying codes of Cartesian product of two graphs. In this paper we consider the identifying codes of lexicographic product G[H] of a connected graph G and an arbitrary graph H, and obtain the minimum…
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…
The three standard products (the Cartesian, the direct and the strong product) of undirected graphs have been wellinvestigated, unique prime factor decomposition (PFD) are known and polynomial time algorithms have been established for…
Recently, we have introduced and modified two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product…
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 $G$.…
It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the…
Recently, we have introduced and modified two graph-decomposition theorems based on a new graph product, motivated by applications in the context of synchronising periodic real-time processes. This vertex-removing synchronised product…
The oriented chromatic number of an oriented graph $\vec G$ is the minimum order of an oriented graph $\vev H$ such that $\vec G$ admits a homomorphism to $\vev H$. The oriented chromatic number of an undirected graph $G$ is then the…
The chromatic number $\chi((G,\sigma))$ of a signed graph $(G,\sigma)$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= \sigma(e) c(w)$ for every edge $e = vw$. Let…
Cartesian products of graphs and hypergraphs have been studied since the 1960s. For (un)directed hypergraphs, unique \emph{prime factor decomposition (PFD)} results with respect to the Cartesian product are known. However, there is still a…
A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a…