R-dynamic Coloring Research Articles

On r-dynamic Coloring of Ladder Graph and Tadpole Graph Using m-shadow Operation

An r-dynamic coloring is a proper k-coloring of a graph G = {V,E} such that the neighbors of every vertex v ∈ V(G) is colored using ς: V(G) → S(c) where S(c) is a set of colors. The coloring is made in such a way that it satisfies the conditions: (i) For any edge uv ∈ E(G), the color of u and color of v are distinct and (ii) The cardinality of coloring the neighbors of any vertex v should be greater than or equal to min{r, d(vG)}, where d(vG) is the degree of the vertex v. In this paper the lower bounds for the r-dynamic coloring of m-shadow graph of ladder graph Dm(Ln) and tadpole graph Dm(Ln,p) are attained. Using the lower bounds the exact solution of r-dynamic chromatic number of the ladder graph Ln and tadpole graph Ln,p by m-shadow operation are obtained.

Research Based Learning and STEM Learning Activities: The Use of R-Dynamic Coloring to Improve the Students Meta-literacy in Solving a Tessellation Decoration Problem


 
 
 The metaliteracy ability is a very important in today's life, especially to live in the disruptive technology era. Metaliteracy is an ability that goes beyond metacognition and technological literacy. However, the students' metalliteracy ability is still relatively low. One of the causes of the low ability is due to the learning model that has been applied so far has not been able to bring out this ability. Therefore, in this study, an RBL learning model that is integrated with the STEM approach will be applied in solving the r-Dynamic coloring problem. By r-dynamic coloring, we mean a proper k-coloring c of G if for every vertex v ∈V (G) satisfies |c(N(v))| ≥ min{r,d(v)}. The minimum k such that G has an r-dynamic coloring with k colors is called an r-dynamic chromatic number, denoted by r(G). By using the r-dynamic coloring technique, we will improve students' metaliteracy in solving the tessellation decoration problem. Therefore, in this research, the syntax of learning activities of the Research-Based Learning and STEM approach will be developed including the assesment indicator of the metaliteracy ability.
 
 

On r-dynamic coloring of central vertex join of path, cycle with certain graphs

Let G = (V, E) be a simple finite connected and undirected graph with n vertices and m edges. The n vertices are assigned the colors through mapping c : V [G] → I +. An r-dynamic coloring is a proper k-coloring of a graph G such that each vertex of G receive colors in at least min{deg(υ),r} different color classes. The minimum k such that the graph G has r-dynamic k coloring is called the r-dynamic chromatic number of graph G denoted as χ r (G). Let G 1 and G 2 be a graphs with n 1 and n 2 vertices and m 1 and m 2 edges. The central vertex join of G1 and G 2 is the graph is obtained from C(G 1) and G 2 joining each vertex of G 1 with every vertex of G 2. The aim of this paper is to obtain the lower bound for r-dynamic chromatic number of central vertex join of path with a graph G, central vertex join of cycle with a graph G and r-dynamic chromatic number of and respectively.

Single Valued Neutrosophic Dynamic Vertex Coloring of Some Cartesian Product and Join of SVNG′s

The concept of neutrosophic sets makes it easier to analyse values that are ambiguous or indeterminate. We illustrate the concept of single-valued neutrosophic R-dynamic vertex coloring in this study. In addition, the single valued neutrosophic dynamic vertex coloring of the Cartesian product of path with cycle, cycle with cycle, complete SVNG with complete SVNG and cycle has been determined. Also, we have discussed the single valued neutrosophic dynamic vertex coloring of join of complete SVNG with complete SVNG and cycle, cycle with cycle.

On r-dynamic chromatic number of some brick product graphs C(2n, 1, p)

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that [Formula: see text] for each vertex [Formula: see text]. The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of brick product graphs.

Design Thinking on δ-Dynamic Coloring of Central Vertex Join of Graphs

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, d(v)}, for each v ∈ V(G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the δ-dynamic chromatic number of the central vertex join of two graphs.

On r-dynamic coloring of comb graphs

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

On the r-dynamic coloring of the direct product of a path and a k-subdivision of a star graph

In this paper, we obtain explicitly the [Formula: see text]-dynamic chromatic number of the direct product of a path [Formula: see text] with a [Formula: see text]-subdivision [Formula: see text] of a star graph, for every positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Particularly, it is obtained that [Formula: see text].

On r-dynamic vertex coloring of some flower graph families

Let [Formula: see text] be a simple, connected undirected graph with [Formula: see text] vertices and [Formula: see text] edges. Let vertex coloring [Formula: see text] of a graph [Formula: see text] be a mapping [Formula: see text], where [Formula: see text] and it is [Formula: see text]-colorable. Vertex coloring is proper if none of the any two neighboring vertices receives the similar color. An [Formula: see text]-dynamic coloring is a proper coloring such that [Formula: see text] min[Formula: see text], for each [Formula: see text][Formula: see text]. The [Formula: see text]-dynamic chromatic number of a graph [Formula: see text] is the minutest coloring [Formula: see text] of [Formula: see text] which is [Formula: see text]-dynamic k-colorable and denoted by [Formula: see text]. By a simple view, we exhibit that [Formula: see text], howbeit [Formula: see text] cannot be arbitrarily small. Thus, finding the result of [Formula: see text] is useful. This study gave the result of [Formula: see text]-dynamic chromatic number for the central graph, Line graph, Subdivision graph, Line of subdivision graph, Splitting graph and Mycielski graph of the Flower graph [Formula: see text] denoted by [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] respectively.

On r− dynamic coloring of the gear graph families

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, d(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r−dynamic chromatic number of the middle, central and line graphs of the gear graph.

On the r-dynamic coloring of the direct product of a path with either a complete graph or a wheel graph

In this paper, it is explicitly determined the <i>r</i>-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper.

Independent Domination Number of Operation Graph

Let G be a simple, undirected and connected graph. An independent set or stable set is a set of vertices in a graph in which no two of vertices are adjacent. A set D of vertices of graph G is called a dominating set if every vertex u ∈ V (G) − D is adjacent to some vertex v ∈ D. A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. A minimum independent dominating set is an independent set of smallest possible size for a given graph G. This size is called the independence number of G, and denoted i(G). Operation Graph is a technical to get a new graph types by performing the operation of two or more graphs. Power Graph is a operation graph where let the graph G and H , notation of the power graph is (GH ). Keywords: r-dynamic coloring, r-dynamic chromatic number, graph operations.

On r-Dynamic Coloring for Graph Operation of Cycle, Star, Complete, and Path

For integer k, r &gt; 0, (k, r) -coloring of graph G is a proper coloring on the vertices of G by k-colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different color. By a proper k -coloring of graph G, we mean a map c : V (G) → S, where |S| = k, such that any two adjacent vertices are different color. An r -dynamic k -coloring is a proper k -coloring c of G such that |c(N (v))| ≥ min{r, d(v)} for each vertex v in V (G), where N (v) is the neighborhood of v and c(S) = {c(v) : v ∈ S} for a vertex subset S . The r-dynamic chromatic number, written as χr (G), is the minimum k such that G has an r-dynamic k-coloring. Note the 1-dynamic chromatic number of graph is equal to its chromatic number, denoted by χ(G), and the 2-dynamic chromatic number of graph denoted by χd (G). By simple observation with a greedy coloring algorithm, it is easy to see that χr (G) ≤ χr+1(G), however χr+1(G) − χr (G) does not always have the same difference. Thus finding an exact values of χr (G) is significantly useful. In this paper, we investigate the some exact value of χr (G) when G is for an operation product of cycle, star, complete, and path graphs.

On r-Dynamic Coloring of Neighborhood Corona of Path with Some graphs

Consider the simple, finite, connected and undirected graph H = (V, E) in which V and E denotes the vertex set and edge set of the graph H. The r-dynamic coloring of a graph H is the proper p-coloring of the vertices of the graph H in which |c(N(a)| ≥ min{r, d(a)}, for each . The lowest p which allows H an r-dynamic coloring with p colors is called the r-dynamic chromatic number of the graph H and it is denoted as X r (H). Let H 1 and H 2 be two graphs with vertex disjoint sets of n 1 and n 2vertices. The neighborhood corona of two graphs H 1 and H 2 is obtained by taking one copy of the graph H 1 and n 1 copies of the graph H 2 and by joining each neighbor of the ith vertex of H 1 to each and every vertex of the ith copy of H 2. It is denoted as . In this paper, we determine the r-dynamic chromatic number of the neighborhood corona of path graph Pm with path Pn , complete graph Kn , cycle Cn and star graph K1,n . These graphs are denoted as and respectively.

R-Dynamic Coloring on Snark Families

An r-dynamic coloring of a graph is a proper minimum coloring of the vertices such that , for each and it is denoted by . Snark are bridgeless cubic connected graph in which every vertex has three neighbors. In this paper, we shown the r-dynamic coloring for Celmins-swart snark, Double star snark, Loupekine snark, Szekeres snark, Watkins snark and infinite family of flower snark with its special case as Tietze’s graph. This result reinforce that most of the snark are 6-colorable for its maximum degree and also we give the procedures to construct a r-dynamic coloring of each snark.

The influence of research-based learning implementation in improving students’ combinatorial thinking skills in solving local irregularity vertex r-dynamic coloring

This research is one of the studies aimed at applying research-based learning and aims to determine its effect on the combinatorial thinking skills of students in solving local irregularity vertex r-dynamic coloring problem. The research method used in this research is mix-method. Mix-method research is a combination of qualitative research and quantitative research. The qualitative method is used to obtain data related to the results of testing on each indicator and the results of the portrait phase while the quantitative method is used for statistical analysis. The experimental class consisted of 41 respondents and the control class consisted of 30 respondents. The treatment given is different, namely between the treatment in the experimental class and the treatment in the control class. In the experimental class applied learning tools that have been developed and applied research-based learning methods. While the control class used research-based learning methods without the learning tools that have been developed. Based on the treatment that has been given, the results of students’ combinatorial thinking skills show for the control class 53% are categorized as low-level, 37% are categorized as a medium level, and 10% are categorized as a high level of combinatorial thinking skills and then experimental class that 34% are categorized as low level, 44% are categorized as a medium level, and 22% are categorized as combinatorial high-level thinking skills. The results of this study of homogeneity of two classes by using a pre-test score of 0,701 result show sig> 0.05 Thus Spake the differences of the mean of two classes is not significant. The inferential statistical result of the independent sample t-test on the posttest results Showed that the sig 2-tailed value was 0.000 (p < 0.05) so that was significant. The conclusion is there is a significant impact of the application of research-based learning in improving the students’ combinatorial thinking skills in solving local irregularity vertex r-dynamic coloring.

On r-dynamic local irregularity vertex coloring of special graphs

In this paper, we study a new notion of graph coloring, that is a local irregularity vertex r-Dynamic coloring. We combine irregular local and the principles of r-dynamic coloring and we assign color to all vertices by using the weights of local irregularity vertex. We define l : V(G) → {1, 2, …, k} as a vertex irregular k-labeling and w : V(G) → N where w(u) = ∑ v∈N(u) l(v). By a local irregularity vertex coloring, we define a condition for f if for every uv ∈ E(G), w(u) ≠ w(v) and max(l) = min{max{li }; li vertex irregular labeling}. Each vertex of the weight as a color should satisfy the r-dynamic condition, namely |w(N(v)) ≥ min{r, d(v)} and each the adjacent vertices must be different. In this paper, we study the local irregularity vertex r-Dynamic coloring of special graph, namely triangular book graph, central of friendship graph, tapol graph, and rectangular book graph.

On r-dynamic vertex coloring of line, middle, total of lobster graph

Let G be a simple, connected and undirected graph that has a set of vertex and edge. The degree of v ∈ V(G) is denoted by d(v). The maximum and minimum degree of G respectively are Δ(G) and δ(G). The r-dynamic color of the graph G is calculated as a map c from V to a color set such that if u, v ∈ V(G) is adjacent, then c(u) ≠ c(v), and for each v ∈ V(G), |c(N(v))| ≥ min{r, d(v)}. The number of r-dynamic coloring of G denoted by χr (G) is minimum color k in G. In this paper, we have obtained the r-dynamic vertex coloring of line, middle, total of lobster graph ℒn (2, 1).

R-Dynamic coloring of the corona product of graphs

Let [Formula: see text] be a graph. A proper [Formula: see text]-coloring of graph [Formula: see text] is [Formula: see text]-dynamic coloring if for every [Formula: see text], the neighbors of vertex [Formula: see text] receive at least min[Formula: see text] different colors. The minimum [Formula: see text] such that graph [Formula: see text] has [Formula: see text]-dynamic [Formula: see text] coloring is called the [Formula: see text]-dynamic chromatic number, denoted by [Formula: see text]. In this paper, we study the [Formula: see text]-dynamic coloring of corona product of graph. The corona product of graph is obtained by taking a number of vertices [Formula: see text] copy of [Formula: see text], and making the [Formula: see text]th of [Formula: see text] adjacent to every vertex of the [Formula: see text]th copy of [Formula: see text]. We obtain the lower bound of [Formula: see text]-dynamic chromatic number of corona product of graphs and some exact value.

The r-dynamic chromatic number of corona of order two of any graphs with complete graph

The corona of two graphs G and H is a graph G ⊙ H formed from one copy of G and |V (G)| copies of H where the ith vertex of G is adjacent to every vertex in the ith copy of H. For any integer l ≤ 2, we define the graph G ⊙l H recursively from G ⊙ H as G ⊙l H = (G ⊙l–1 H) ⊙ H. Graph G ⊙l H is also named as l-corona product of G and H. The r-dynamic chromatic number is the minimum k such that graph G has an r-dynamic k-coloring, denoted by χr(G). An r-dynamic coloring of a graph G is a proper k-coloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. In this paper, we analyze the r-dynamic chromatic number of corona order two graph, denoted by G ⊙2 H; G is complete graph and H are any graph.