Minimum Number Of Colors Research Articles

BILANGAN KETERHUBUNGAN TITIK PELANGI BEBERAPA KELAS GRAF

A graph G is called a rainbow vertex connected if every two vertices G are connected by a rainbow path, that is, a path whose all the internal vertices are of a different color. The rainbow vertex connection number of graph G denoted by rvc(G) is the minimum number of colors used to color all vertices by G such that the graph G is connected to rainbow vertex. The rainbow vertex connection number in a graph will not be less than the diameter of the graph minus one. The rainbow vertex connection number discussed in this article for various classes of graphs include complete graph Kn, complete bipartite graph Km,n , wheel graph Wn , two-layer wheel graph Wn2, complete multipartite graph Kn1,n2,...,nt , path Pn, comb graph GSn, graph , graph , graph , graph .
 Keywords: graph, vertex coloring, rainbow vertex connection number.

Total Rainbow Connection Number Of Shackle Product Of Antiprism Graph (〖AP〗_3)

Function if is said to be k total rainbows in , for each pair of vertex there is a path called with each edge and each vertex on the path will have a different color. The total connection number is denoted by trc defined as the minimum number of colors needed to make graph to be total rainbow connected. Total rainbow connection numbers can also be applied to graphs that are the result of operations. The denoted shackle graph is a graph resulting from the denoted graph where t is number of copies of G. This research discusses rainbow connection numbers rc and total rainbow connection trc(G) using the shackle operation, where is the antiprism graph . Based on this research, rainbow connection numbers rc shack , and total rainbow connection trc shack for .

On tangencies among planar curves with an application to coloring L-shapes

We prove that there are O(n) tangencies among any set of n red and blue planar curves in which every pair of curves intersects at most once and no two curves of the same color intersect. If every pair of curves may intersect more than once, then it is known that the number of tangencies could be super-linear. However, we show that a linear upper bound still holds if we replace tangencies by pairwise disjoint connecting curves that all intersect a certain face of the arrangement of red and blue curves.The latter result has an application for the following problem studied by Keller, Rok and Smorodinsky [Disc. Comput. Geom. (2020)] in the context of conflict-free coloring of string graphs: what is the minimum number of colors that is always sufficient to color the members of any family of ngrounded L-shapes such that among the L-shapes intersected by any L-shape there is one with a unique color? They showed that O(log3n) colors are always sufficient and that Ω(logn) colors are sometimes necessary. We improve their upper bound to O(log2n).

Equitable Edge Coloring of Splitting Graph of Some Classes of Wheel Graphs

The coloring of all the edges of a graph G with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of W n , D W n and G n by determining its edge chromatic number.

Equitable Edge Coloring of Splitting Graph of Some Classes of Wheel Graphs

The coloring of all the edges of a graph \(G\) with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of \(W_n\), \(DW_n\) and \(G_n\) by determining its edge chromatic number.

A Triangle-free, 4-chromatic [formula omitted] Euclidean distance graph Scavenger hunt!

For d>0, define G(Q3,d) to be the graph whose set of vertices is the rational space Q3, where two vertices are adjacent if and only if they are a Euclidean distance d apart. Let χ(Q3,d) be the chromatic number of such a graph or, in other words, the minimum number of colors needed to color the points of Q3 so that no two points at distance d apart receive the same color. An open problem, originally posed by Benda and Perles in the 1970s, asks if there exists d such that χ(Q3,d)=3. Through numerous efforts over the years, χ(Q3,d) has been determined for many values of d, and for all those distances d where χ(Q3,d) has not been exactly pinned down, it is known that χ(Q3,d)∈{3,4}. In our work, we detail several search algorithms we have employed to find 4-chromatic subgraphs of various graphs G(Q3,d) whose chromatic number was previously unknown. Ultimately, we conjecture that no 3-chromatic G(Q3,d) exists. Along the way, we pose a few related questions that we feel are of interest in their own right.

Loose Edge-Connection of Graphs

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and obtained a lot of attention. In this paper, we investigate the loose edge-connection of graphs. A connected edge-coloured graph G is loose edge-connected if between any two of its vertices there is a path of length one, or a bi-coloured path of length two, or a path of length at least three with at least three colours used on its edges. The minimum number of colours, used in a loose edge-colouring of G, is called the loose edge-connection number and denoted {{,textrm{lec},}}(G). We determine the precise value of this parameter for any simple graph G of diameter at least 3. We show that deciding, whether {{,textrm{lec},}}(G) = 2 for graphs G of diameter 2, is an NP-complete problem. Furthermore, we characterize all complete bipartite graphs K_{r,s} with {{,textrm{lec},}}(K_{r,s}) = 2.

Two-distance vertex-distinguishing total coloring of subcubic graphs

A 2-distance vertex-distinguishing total coloring of graph G is a proper total coloring of G such that any pair of vertices at distance of two have distinct sets of colors. The 2-distance vertex-distinguishing total chromatic number $\chi_{d2}^{''}(G)$ of G is the minimum number of colors needed for a 2-distance vertex-distinguishing total coloring of G. In this paper, it's proved that if G is a subcubic graph, then $\chi_{d2}^{''}(G)\le 7$.

BILANGAN KROMATIK EQUITABLE PADA GRAF BINTANG, GRAF LOLIPOP, DAN GRAF PERSAHABATAN

Let G be a connected and undirected graph. Vertex coloring in a graph G is a mapping from the set of vertices in G to the set of colors such that every two adjacent vertices have different colors. There are many types of vertex coloring, such as complete coloring, k-differential coloring, and equitable coloring. Equitable coloring of G is a vertex coloring of G that satisfies the condition that for each induced color class it has an equitable cardinality with difference 0 or 1. The minimum number of colors used for such coloring of G is called the equitable chromatic number of G, denoted by χe(G). In this study, we only concern with graphs that have a central vertex, which means a vertex that is adjacent to every other vertex, in particular on the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). This research aims to formulate the equitable chromatic number of the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). The first step taken in this research is to apply vertex coloring to Sn, Ln, and fn. After that, the color classes of the vertex set are obtained and its cardinality is determined. Next, analyze that the applied vertex coloring meets the definition of equitable coloring. Then, prove that the number of colors used is minimum. Thus, the chromatic number for each graph is obtained and proved. Based on this research, the equitable chromatic number of Sn is ⌈n/2⌉ + 1, the equitable chromatic number of Ln is n, and the equitable chromatic number of fn is 3, for n = 1 and n + 1, for n ≥ 2.

Brooks-type theorem for r-hued coloring of graphs

An r-hued coloring of a simple graph G is a proper coloring of its vertices such that every vertex v is adjacent to at least min{r,deg(v)} differently colored vertices. The minimum number of colors needed for an r-hued coloring of a graph G, the r-hued chromatic number, is denoted by χr(G). In this note we show that χr(G)≤(r−1)(Δ(G)+1)+2, for every simple graph G and every r≥2, which in the case when r<Δ(G) improves the presently known Δ(G)-based upper bound on χr(G), namely rΔ(G)+1. We also discuss the existence of graphs whose r-hued chromatic number is close to (r−1)(Δ+1)+2 and we prove that there is a bipartite graph of maximum degree Δ whose r-hued chromatic number is (r−1)Δ+1 for every r≥2 and infinitely many values of Δ≥r+2; we believe that (r−1)Δ(G)+1 is the best upper bound on the r-hued chromatic number of any bipartite graph G.

Total coloring conjecture on certain classes of product graphs

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph G , denoted by χ ″( G ) , is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any graph G , Δ ( G )+1 ≤ χ ″( G )≤ Δ ( G )+2 , where Δ ( G ) is the maximum degree of G . In this paper, we prove the Behzad and Vizing conjecture for Indu - Bala product graph, Skew and Converse Skew product graph, Cover product graph, Clique cover product graph and Comb product graph.

On the inverse graph of a finite group and its rainbow connection number

A rainbow path in an edge-colored graph G is a path that every two edges have different colors. The minimum number of colors needed to color the edges of G such that every two distinct vertices are connected by a rainbow path is called the rainbow connection number of G . Let ( Γ , *) be a finite group with T Γ = { t ∈ Γ | t ≠ t −1 } . The inverse graph of Γ , denoted by I G ( Γ ) , is a graph whose vertex set is Γ and two distinct vertices, u and v , are adjacent if u * v ∈ T Γ or v * u ∈ T Γ . In this paper, we determine the necessary and sufficient conditions for the inverse graph of a finite group to be connected. We show that the inverse graph of a finite group is connected if and only if the group has a set of generators whose all elements are non-self-invertible. We also determine the rainbow connection numbers of the inverse graphs of finite groups.

1-planar graphs are odd 13-colorable

An odd coloring of a graph G is a proper coloring such that any non-isolated vertex in G has a color appearing an odd number of times on its neighbors. The odd chromatic number, denoted by χo(G), is the minimum number of colors that admits an odd coloring of G. Petruševski and Škrekovski in 2021 introduced this notion and proved that if G is planar, then χo(G)≤9 and conjectured that χo(G)≤5. More recently, Petr and Portier improved 9 to 8. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showed that every 1-planar graph is odd 23-colorable. In this paper, we improved this result and showed that every 1-planar graph is odd 13-colorable.

The mod k $k$ chromatic index of random graphs

AbstractThe mod chromatic index of a graph is the minimum number of colors needed to color the edges of in a way that the subgraph spanned by the edges of each color has all degrees congruent to . Recently, the authors proved that the mod chromatic index of every graph is at most , improving, for large , a result of Scott. Here we study the mod chromatic index of random graphs. We prove that for every integer , there is such that if and as , then the following holds: if is odd, then the mod chromatic index of is asymptotically almost surely (a.a.s.) equal to , while if is even, then the mod chromatic index of (respectively, ) is a.a.s. equal to (respectively, ).

Robust Coloring Optimization Model on Electricity Circuit Problems

The Graph Coloring Problem (GCP) is assigning different colors to certain elements in a graph based on certain constraints and using a minimum number of colors. GCP can be drawn into optimization problems, namely the problem of minimizing the color used together with the uncertainty in using the color used, so it can be assumed that there is an uncertainty in the number of colored vertices. One of the mathematical optimization techniques in dealing with uncertainty is Robust Optimization (RO) combined with computational tools. This article describes a robust GCP using the Polyhedral Uncertainty Theorem and model validation for electrical circuit problems. The form of an electrical circuit color chart consists of corners (components) and edges (wires or conductors). The results obtained are up to 3 colors for the optimization model for graph coloring problems and up to 5 colors for robust optimization models for graph coloring problems. The results obtained with robust optimization show more colors because the results contain uncertainty. When RO GCP is applied to an electrical circuit, the model is used to place the electrical components in the correct path so that the electrical components do not collide with each other.

Lower bounds on the Erdős–Gyárfás problem via color energy graphs

AbstractGiven positive integers and , a ‐coloring of the complete graph is an edge‐coloring in which every ‐clique receives at least colors. Erdős and Shelah posed the question of determining , the minimum number of colors needed for a ‐coloring of . In this paper, we expand on the color energy technique introduced by Pohoata and Sheffer to prove new lower bounds on this function, making explicit the connection between bounds on extremal numbers and . Using results on the extremal numbers of subdivided complete graphs, theta graphs, and subdivided complete bipartite graphs, we generalize results of Fish, Pohoata, and Sheffer, giving the first nontrivial lower bounds on for some pairs and improving previous lower bounds for other pairs.

Approximation algorithm for minimum q-dominator partization problem

A dominator coloring of a graph [Formula: see text] is a proper vertex coloring with a domination property that each vertex dominates a color class. The minimum number of colors used in a dominator coloring of [Formula: see text] is called dominator chromatic number of [Formula: see text] and is denoted as [Formula: see text]. Graphs with [Formula: see text] have characterizations and such graphs can be recognized in polynomial time. However, for [Formula: see text], deciding whether [Formula: see text] is NP-hard. In this paper, we investigate the computational complexity of minimum [Formula: see text]-dominator partization problem (Min-[Formula: see text]-D-Partz). In Min-[Formula: see text]-D-Partz, given a graph [Formula: see text], the objective is to find a vertex set [Formula: see text] of minimum size such that [Formula: see text]. We prove that Min-[Formula: see text]-D-Partz is APX-complete and has a two-factor approximation algorithm. We also prove that Min-[Formula: see text]-D-Partz cannot be approximated within any constant factor and can be approximated within a factor of [Formula: see text].

Decremental Optimization of Vertex-Coloring Under the Reconfiguration Framework

Suppose that we are given a positive integer k, and a k-(vertex-)coloring of a given graph G. Then we are asked to find a coloring of G using the minimum number of colors among colorings that are reachable from by iteratively changing a color assignment of exactly one vertex while maintaining the property of k-colorings. In this paper, we give linear-time algorithms to solve the problem for graphs of degeneracy at most two and for the case where . These results imply linear-time algorithms for series-parallel graphs and grid graphs. In addition, we give linear-time algorithms for chordal graphs and cographs. On the other hand, we show that, for any , this problem remains NP-hard for planar graphs with degeneracy three and maximum degree four. Thus, we obtain a complexity dichotomy for this problem with respect to the degeneracy of a graph and the number k of colors.

The Locating Chromatic Number for Certain Operation of Origami Graphs

The locating chromatic number introduced by Chartrand et al. in 2002 is the marriage of the partition dimension and graph coloring. The locating chromatic number depends on the minimum number of colors used in the locating coloring and the different color codes in vertices on the graph. There is no algorithm or theorem to determine the locating chromatic number of any graph carried out for each graph class or the resulting graph operation. This research is the development of scientific theory with a focus of the study on developing new ideas to determine the extent to which the locating chromatic number of a graph increases when applied to other operations. The locating chromatic number of the origami graph was obtained. The next exciting thing to know is locating chromatic number for certain operation of origami graphs. This paper discusses locating chromatic number for specific operation of origami graphs. The method used in this study is to determine the upper and lower bound of the locating chromatic number for certain operation of origami graphs. The result obtained is an increase of one color in the locating chromatic number of origami graphs.

Brooks' theorem for 2-fold coloring

The two-fold chromatic number of a graph is the minimum number of colors needed to ensure that there is a way to color the graph so that each vertex gets two distinct colors, and adjacent vertices have no colors in common. The Ore degree is the maximum sum of degrees of an edge in a graph. We prove that, for 2-connected graphs, the two-fold chromatic number is at most the Ore degree, unless G is a complete graph or an odd cycle.