Chromatic Number Of Cartesian Product Research Articles

Results on Grundy Chromatic Number of Join Graph of Graphs

A Grundy k -coloring of a graph G is a proper k -coloring of vertices in G using colors { 1 , 2 , ⋯ , k } such that for any two colors x and y , x < y , any vertex colored y is adjacent to some vertex colored x . The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph G , denoted by Γ ( G ) , is the largest integer k , such that there exists a Grundy k -coloring for G . It can be easily seen that Γ ( G ) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of G . In this paper, we obtain the Grundy chromatic number of Cartesian Product of path graph, complete graph, cycle graph, complete graph, wheel graph and star graph.

On r-hued coloring of product graphs

A (k,r)-coloring of a graph G is a proper coloring with k colors such that for every vertex v with degree d(v) in G, the color number of the neighbors of v is at least min{d(v),r}. The smallest integer k such that G has a (k,r)-coloring is called the r-hued chromatic number and denoted by χr(G). In Kaliraj et al. [Taibah Univ. Sci. 14 (2020) 168–171], it is determined the 2-hued chromatic numbers of Cartesian product of complete graph and star graph. In this paper, we extend its result and determine the r-hued chromatic number of Cartesian product of complete graph and star graph.

On construction of fuzzy chromatic number of cartesian product of path and other fuzzy graphs

On construction of fuzzy chromatic number of cartesian product of path and other fuzzy graphs

Algorithms to determine fuzzy chromatic number of cartesian product and join of fuzzy graphs

Throughout this paper, we use the concept of fuzzy chromatic number (FCN) of fuzzy graphs based on δ-fuzzy independent vertex sets with δ ∈ [0,1] published in 2015. In this paper, we construct two type algorithms to determine fuzzy chromatic number of cartesian product and join of fuzzy graphs. The first type algorithm is constructed to find FCN of specific fuzzy graphs based on formulas in the theorems. The second type algorithm is designed to establish FCN of cartesian product and join of any fuzzy graphs based on the concepts of the cartesian product and join of fuzzy graphs and fuzzy chromatic algorithm. Experimental results of the algorithms are also discussed.

On dynamic colouring of cartesian product of complete graph with some graphs

ABSTRACT A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by . We denote the cartesian product of G and H by . In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph , complete graph with complete bipartite graph and wheel graph with complete graph .

On Neighbor chromatic number of grid and torus graphs

Abstract A set S ⊆ V is a neighborhood set of a graph G = (V, E), if G = ∪ v∈S 〈 N[v] 〉, where 〈 N[v] 〉 is the subgraph of a graph G induced by v and all vertices adjacent to v. A neighborhood set S is said to be a neighbor coloring set if it contains at least one vertex from each color class of a graph G, where color class of a colored graph is the set of vertices having one particular color. The neighbor chromatic number χ n (G) is the minimum cardinality of a neighbor coloring set of a graph G. In this article, some results on neighbor chromatic number of Cartesian products of two paths (grid graph) and cycles (torus graphs) are explored.

On the Alon-Tarsi Number and Chromatic-Choosability of Cartesian Products of Graphs

We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $\chi(G) \leq \chi_\ell(G) \leq \chi_p(G) \leq AT(G)$, where  $\chi(G)$ is the chromatic number, $\chi_\ell(G)$ is the list chromatic number, and $\chi_p(G)$ is the paint number of $G$. In this paper we find families of graphs $G$ and $H$ such that $\chi(G \square H) = AT(G \square H)$, reducing this sequence of inequalities to equality.
 We show that the Alon-Tarsi number of the Cartesian product of an odd cycle and a path is always equal to 3. This result is then extended to show that if $G$ is an odd cycle or a complete graph and $H$ is a graph on at least two vertices containing the Hamilton path $w_1, w_2, \ldots, w_n$ such that for each $i$, $w_i$ has a most $k$ neighbors among $w_1, w_2, \ldots, w_{i-1}$, then $AT(G \square H) \leq \Delta(G)+k$ where $\Delta(G)$ is the maximum degree of $G$.  We discuss other extensions for $G \square H$, where $G$ is such that $V(G)$ can be partitioned into odd cycles and complete graphs, and $H$ is a graph containing a Hamiltonian path. We apply these bounds to get chromatic-choosable Cartesian products, in fact we show that these families of graphs have $\chi(G) = AT(G)$, improving previously known bounds.

Oriented chromatic number of Cartesian products and strong products of paths

Oriented chromatic number of Cartesian products and strong products of paths

The k-Distance Independence Number and 2-Distance Chromatic Number of Cartesian Products of Cycles

A set $$S \subseteq V(G)$$ is a k-distance independent set of a graph G if the distance between every two vertices of S is greater than k. The k-distance independence number $$\alpha _k(G)$$ of G is the maximum cardinality over all k-distance independent sets in G. A k-distance coloring of G is a function f from V(G) onto a set of positive integers (colors) such that for any two distinct vertices u and v at distance less than or equal to k we have $$f(u) \not = f(v)$$ . The k-distance chromatic number of a graph G is the smallest number of colors needed to have a k-distance coloring of G. The k-distance independence numbers and 2-distance chromatic numbers of Cartesian products of cycles are considered. A computer-aided method with the isomorphic rejection is used to determine the k-distance independence numbers of Cartesian products of cycles. By using these results, several lower and upper bounds on the maximal cardinality $$A^L_q(n, d)$$ of a q-ary Lee code of length n with a minimum distance at least d are improved. It is also established that the 2-distance chromatic number of G equals $$\left\lceil \frac{|V(G)|}{\alpha (G^2)} \right\rceil $$ for $$G=C_m \Box C_n \Box C_k$$ , whenever $$k \ge 3$$ and $$(m,n)\in \{(3,3), (3,4), (3,5), (4,4)\}$$ or k, m and n are all multiples of seven. Moreover, it is shown that the 2-distance chromatic number of the three-dimensional square lattice is equal to seven.

The Mediator Chromatic Number of Grid Graphs

In this paper, we determine the mediator chromatic number of cartesian product of a path with and thus prove that the mediator chromatic number of Ladder graphs and Grid graphs are polynomially solvable.

Distinguishing Chromatic Number of Cartesian Products of Graphs

The distinguishing chromatic number $\chi_{_D}(G)$ of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number $\chi(\cdot)$. Our main result is that for every graph G, there exists a constant $d_G$ such that for all $d\geq d_G$ the distinguishing chromatic number of $G^d$ is at most $\chi(G) +1$, where $G^d$ is the Cartesian product of d copies of G. We also prove that for $d\geq5$, the Cartesian product of d complete graphs has distinguishing chromatic number at most one more than the corresponding chromatic number, and we determine the distinguishing chromatic number of hypercubes exactly.

The distinguishing chromatic number of Cartesian products of two complete graphs

A labeling of a graph G is distinguishing if it is only preserved by the trivial automorphism of G . The distinguishing chromatic number of G is the smallest integer k such that G has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product K k □ K n is determined for all k and n . In most of the cases it is equal to the chromatic number, thus answering a question of Choi, Hartke and Kaul whether there are some other graphs for which this equality holds.

On the Packing Chromatic Number of Trees, Cartesian Products and Some Infinite Graphs

Abstract A subset A of vertices in a graph G is an r-packing if the distance between every pair of distinct vertices in A is more than r. The packing chromatic number, χ ρ ( G ) , is the smallest k for which there exists some partition V 1 , V 2 , … , V k of the vertex set of G such that V r is an r-packing. We obtain lower and upper bounds for the packing chromatic number of Cartesian products and subdivisions of finite graphs and study the existence of monotone colorings in trees. The infinite, planar triangular lattice and the three dimensional square lattice are shown to have unbounded packing chromatic number.

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into k packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χ ρ ( G ) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.