Minimum Number Of Colors Research Articles

Perfect Resolution of Strong Conflict-Free Colouring of Interval Hypergraphs

The \(k\) -Strong Conflict-Free ( \(k\) -SCF, in short) colouring problem seeks to find a colouring of the vertices of a hypergraph \(H\) using minimum number of colours so that in every hyperedge \(e\) of \(H\) , there are at least \(\min\{|e|,k\}\) vertices whose colours are different from that of all other vertices in \(e\) . In the case of interval hypergraphs, we present an exact \({\mathsf{P}}\) -time algorithm for the \(k\) -SCF problem thus solving an open problem posed by Cheilaris et al. (2014). We achieve our results by showing that for any hypergraph, a \(k\) -SCF colouring is a proper colouring of a related simple graph which we refer to as a co-occurrence graph . We then show that a co-occurrence graph is obtained by identifying an induced subgraph of a second simple graph that we introduce, which we refer to as the conflict graph . For interval hypergraphs, we show that each co-occurrence graph and the conflict graph are perfect graphs. This property plays a crucial role in our polynomial time algorithm. Secondly, we show that for an interval hypergraph, the \(1\) -SCF colouring number is the minimum partition of its intervals into sets such that each set has an exact hitting set (a hitting set in which each interval is hit exactly once).

RAINBOW VERTEX CONNECTION NUMBER OF BULL GRAPH, NET GRAPH, TRIANGULAR LADDER GRAPH, AND COMPOSITION GRAPH (P_n [P_1

The rainbow connection was first introduced by Chartrand in 2006 and then in 2009 Krivelevich and Yuster first time introduced the rainbow vertex connection. Let graph be a connected graph. The rainbow vertex-connection is the assignment of color to the vertices of a graph , if every vertex on the graph is connected by a path graph that has interior vertices in different colors. The minimum number of colors from the rainbow vertex coloring in the graph is called rainbow vertex connection number which is denoted . The results of the research are the rainbow vertex connection number of bull graph, net graph, triangular ladder graph, and graph composition (Pn[P1]).

On the minimum number of arcs in 4‐dicritical oriented graphs

AbstractThe dichromatic number of a digraph is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph is ‐dicritical if and each proper subdigraph of satisfies . For integers and , we define (resp., ) as the minimum number of arcs possible in a ‐dicritical digraph (resp., oriented graph). Kostochka and Stiebitz have shown that . They also conjectured that there is a constant such that for and large enough. This conjecture is known to be true for . In this work, we prove that every 4‐dicritical oriented graph on vertices has at least arcs, showing the conjecture for . We also characterise exactly the 4‐dicritical digraphs on vertices with exactly arcs.

Graphs with large (1,2)-rainbow connection numbers

Let G be an edge-colored graph. If every subpath of length at most l+1 within a path P in a graph G consists of uniquely colored edges, then P is called an l-rainbow path. A connected graph G is deemed (1,2)-rainbow connected if there exists at least one 2-rainbow path connecting two distinct vertices within G. The minimum number of colors needed to attain (1,2)-rainbow connectedness in a connected graph G, represented as rc1,2(G), is referred to as the (1,2)-rainbow connection number. If G is a nontrivial connected graph of size m, then rc1,2(G)=m if and only if G is the star or double star of size m. Our main goal is to identify all connected graphs G of size m that satisfy the condition m−3≤rc1,2(G)≤m−1.

Local distance antimagic cromatic number of join product of graphs with cycles or paths

Let $G$ be a graph of order $p$ without isolated vertices. A bijection $f: V \to \{1,2,3,\dots,p\}$ is called a local distance antimagic labeling, if $w_f(u)\ne w_f(v)$ for every edge $uv$ of $G$, where $w_f(u)=\sum_{x\epsilon N(u)} {f(x)}$. The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings of $G$. In this paper, we determined the local distance antimagic chromatic number of some cycles, paths, disjoint union of 3-paths. We also determined the local distance antimagic chromatic number of join products of some graphs with cycles or paths.

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

Chromatic numbers of certain hypercube variants

The total (vertex/edge) chromatic number of a graph G, χ ″ ( G ) ( χ ( G ) / χ ′ ( G ) ) is defined to be the minimum number of colours needed to colour the vertices and edges, together termed as elements (vertices/edges) of a graph in such a way that no two adjacent/incident elements (vertices/edges) are given the same colour. In this article, we determine the vertex, edge and total chromatic number of some hypercube variants such as folded hypercube, enhanced hypercube, augmented hypercube and exchanged hypercube.

Locating Chromatic Number of Middle Graph of Path, Cycle, Star, Wheel, Gear and Helm Graphs

Let \(c\) be a proper \(k\)-coloring of a connected graph \(G\) and \(\pi=\{S_{1},S_{2},\ldots,S_{k}\}\) be an ordered partition of the vertex set \(V(G)\) into the resulting color classes, where \(S_{i}\) is the set of all vertices that receive the color \(i\). For a vertex \(v\) of \(G\), the color code \(c_{\pi}(v)\) of \(v\) with respect to \(\pi\) is the ordered \(k\)-tuple \(c_{\pi}(v)=(d(v,S_{1}),d(v,S_{2}),\ldots,d(v,S_{k}))\), where \(d(v,S_{i})=min\{d(v,u):\textit{ } u\in S_{i}\}\) for \(1\leqslant i \leqslant k\). If all distinct vertices of \(G\) have different color codes, then \(c\) is called a locating coloring of \(G\). The locating chromatic number is the minimum number of colors needed in a locating coloring. In this paper, we determine the locating-chromatic number for the middle graphs of Path, Cycle, Wheel, Star, Gear and Helm graphs.

First Fit Algorithm: A Graph Coloring Approach to Conflict-Free University Course Timetabling

Aims: Tackling scheduling issues with the most optimal graph coloring algorithm has consistently posed significant difficulties. The university scheduling problem can be expressed as a graph coloring problem, where courses are depicted as vertices and the connections between courses that have common students or teachers are represented as edges. Subsequently, the task at hand is to assign the vertices with the minimum number of colors. In order to accomplish this task, this paper present a graph coloring technique to conflict free university course timetabling using first fit algorithms.
 Methodology: The conflict graph is partitioned into a set of independent color classes to be assigned time slots and transformed into a conflict-free timetable. The Ladoke Akintola University of Technology (LAUTECH) University Course Timetabling Data was adopted. The allocation of venue based on the allotted time slots is done using first fit packing algorithm. The proposed model is implemented using Python programming language. The developed model had courses being represented as vertices and edges. The course conflict graph was created based on the acquired dataset using vertices-edges relationship diagram. The implemented model is evaluated in terms of Halstead complexity metrics: Program Volume (PV), Program Length (PL), Program Effort (PE), Program Difficulty (PD) and Execution Time (ET). The PV, PL, PE, PD and ET values obtained for the implemented model are 18.45kbits, 0.51, 1037684, 1.97 and 20.45 secs, respectively.
 Conclusion: The proposed model shows a significant improvement over the existing models by producing conflict-free course timetabling problem with better evaluation results. This work will be highly useful in solving various scheduling, optimization and NP-hard related computational problems.

Low-Cost Optical Sensors for Soil Composition Monitoring.

Studying soil composition is vital for agricultural and edaphology disciplines. Presently, colorimetry serves as a prevalent method for the on-site visual examination of soil characteristics. However, this technique necessitates the laboratory-based analysis of extracted soil fragments by skilled personnel, leading to substantial time and resource consumption. Contrastingly, sensor techniques effectively gather environmental data, though they mostly lack in situ studies. Despite this, sensors offer substantial on-site data generation potential in a non-invasive manner and can be included in wireless sensor networks. Therefore, the aim of the paper is to develop a low-cost red, green, and blue (RGB)-based sensor system capable of detecting changes in the composition of the soil. The proposed sensor system was found to be effective when the sample materials, including salt, sand, and nitro phosphate, were determined under eight different RGB lights. Statistical analyses showed that each material could be classified with significant differences based on specific light variations. The results from a discriminant analysis documented the 100% prediction accuracy of the system. In order to use the minimum number of colors, all the possible color combinations were evaluated. Consequently, a combination of six colors for salt and nitro phosphate successfully classified the materials, whereas all the eight colors were found to be effective for classifying sand samples. The proposed low-cost RGB sensor system provides an economically viable and easily accessible solution for soil classification.

2-tone coloring of cactus graphs

A 2-tone coloring of a graph assigns two distinct colors to each vertex with the restriction that adjacent vertices have no common colors, and vertices at distance two have at most one common color. The 2-tone chromatic number of a graph is the minimum number of colors in any 2-tone coloring. A cactus graph has every block a cycle or edge. We determine the 2-tone chromatic number of all cactus graphs with maximum degree Δ≠6. When Δ=6, we determine the 2-tone chromatic number of all cactus graphs that are triangle-free and those with circumference at most 8.

The number of distinguishing colorings of a Cartesian product graph

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold θ(G) of a graph G is the minimum number of colors k required that any arbitrary k-coloring of G is distinguishing. In this paper, we calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.

Coloring ([formula omitted], kite)-free graphs with small cliques

Let Pn and Kn denote the induced path and complete graph on n vertices, respectively. The kite is the graph obtained from a P4 by adding a vertex and making it adjacent to all vertices in the P4 except one vertex with degree 1. A graph is (P5, kite)-free if it has no induced subgraph isomorphic to a P5 or a kite. For a graph G, the chromatic number of G (denoted by χ(G)) is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices receive the same color, and the clique number of G is the size of a largest clique in G. Here, we are interested in coloring the class of (P5, kite)-free graphs with small clique number. It is known that every (P5, kite, K3)-free graph G satisfies χ(G)≤3, every (P5, kite, K4)-free graph G satisfies χ(G)≤4, and that every (P5, kite, K5)-free graph G satisfies χ(G)≤6. In this paper, we show the following: •Every (P5, kite, K6)-free graph G satisfies χ(G)≤7.•Every (P5, kite, K7)-free graph G satisfies χ(G)≤9. We also give examples to show that the above bounds are tight. Based on these partial results and some other examples, we conjecture that every (P5, kite)-free graph G satisfies χ(G)≤⌊3ω(G)2⌋, and that the bound is tight.

A versatile classification tool for galactic activity using optical and infrared colors

Context.The overwhelming majority of diagnostic tools for galactic activity are focused mainly on the classes of active galaxies. Passive or dormant galaxies are often excluded from these diagnostics, which usually employ emission-line features (e.g., forbidden emission lines). Thus, most of them focus on specific types of activity or only on one activity class, for example active galactic nucleus (AGN) galaxiesAims.In this work we used infrared and optical colors to build an all-inclusive galactic activity diagnostic tool that can discriminate between star-forming, AGN, low-ionization nuclear emission-line region, composite, and passive galaxies, and which can be used in local and low-redshift galaxies.Methods.We used the random forest algorithm to define a new activity diagnostic tool. As the ground truth for the training of the algorithm, we considered galaxies that have been classified based on their optical spectral lines. We explored classification criteria based on infrared colors from the first three WISE bands (bands 1, 2, and 3) supplemented with optical colors from theu, g, andrSDSS bands. From them, we sought the combination with the minimum number of colors that provides optimal results. Furthermore, to mitigate biases related to aperture effects, we introduced a new WISE photometric scheme that combines apertures of different sizes.Results.Using machine learning methods, we developed a diagnostic tool that accommodates both active and passive galaxies under one unified classification scheme using just three colors. We find that the combination ofW1-W2, W2-W3, and g-r colors offers a good performance, while the broad availability of these colors for a large number of galaxies ensures it can be applied to large galaxy samples. The overall accuracy is ~81%, and the achieved completeness for each class is ~81% for star-forming, ~56% for AGN, ~68% for LINER, ~65% for composite, and ~85% for passive galaxies.Conclusions.Our diagnostic represents a significant improvement over existing infrared diagnostics because it includes all types of active galaxies, as well as passive galaxies, extending their application to the local Universe. The inclusion of the optical colors improves its ability to identify low-luminosity AGN galaxies, which are generally confused with star-forming galaxies, and helps us identify cases of starbursts with extreme mid-infrared colors that mimic obscured AGN galaxies, a well-known problem for most infrared diagnostics.

On local distance antimagic labeling of graphs

Let G = ( V , E ) be a graph of order n and let f : V → { 1 , 2 … , n } be a bijection. For every vertex v ∈ V , we define the weight of the vertex v as w ( v ) = ∑ x ∈ N ( v ) f ( x ) where N(v) is the open neighborhood of the vertex v. The bijection f is said to be a local distance antimagic labeling of G if w ( u ) ≠ w ( v ) for every pair of adjacent vertices u , v ∈ V . The local distance antimagic labeling f defines a proper vertex coloring of the graph G, where the vertex v is assigned the color w(v). We define the local distance antimagic chromatic number χ l d ( G ) to be the minimum number of colors taken over all colorings induced by local distance antimagic labelings of G. In this paper we obtain the local distance antimagic labelings for several families of graphs including the path Pn , the cycle Cn , the wheel graph Wn , friendship graph Fn , the corona product of graphs G ° K m ¯ , complete multipartite graph and some special types of the caterpillars. We also find upper bounds for the local distance antimagic chromatic number for these families of graphs.

ON RAINBOW ANTIMAGIC COLORING OF SNAIL GRAPH(S_n ), COCONUT ROOT GRAPH (Cr_(n,m) ), FAN STALK GRAPH (Kt_n ) AND THE LOTUS GRAPH(Lo_n )

Rainbow antimagic coloring is a combination of antimagic labeling and rainbow coloring. Antimagic labeling is labeling of each vertex of the graph with a different label, so that each the sum of the vertices in the graph has a different weight. Rainbow coloring is part of the rainbow-connected edge coloring, where each graph has a rainbow path. A rainbow path in a graph is formed if two vertices on the graph do not have the same color. If the given color on each edge is different, for example in the function it is colored with a weight , it is called rainbow antimagic coloring. Rainbow antimagic coloring has a condition that every two vertices on a graph cannot have the same rainbow path. The minimum number of colors from rainbow antimagic coloring is called the rainbow antimagic connection number, denoted by In this study, we analyze the rainbow antimagic connection number of snail graph , coconut root graph , fan stalk graph and lotus graph .

Odd-sum colorings of planar graphs

A coloring of a graph G is a map f:V(G)→Z+ such that f(v)≠f(w) for all vw∈E(G). A coloring f is an odd-sum coloring if ∑w∈N[v]f(w) is odd, for each vertex v∈V(G). The odd-sum chromatic number of a graph G, denoted χos(G), is the minimum number of colors used (that is, the minimum size of the range) in an odd-sum coloring of G. Caro, Petruševski, and Škrekovski showed, among other results, that χos(G) is well-defined for every finite graph G and , in fact, χos(G)≤2χ(G). Thus, χos(G)≤8 for every planar graph G (by the 4 Color Theorem), χos(G)≤6 for every triangle-free planar graph G (by Grötzsch’s Theorem), and χos(G)≤4 for every bipartite graph.Caro et al. asked, for every even Δ≥4, whether there exists gΔ such that if G is planar with maximum degree Δ and girth at least gΔ then χos(G)≤5. They also asked, for every even Δ≥4, whether there exists gΔ such that if G is planar and bipartite with maximum degree Δ and girth at least gΔ then χos(G)≤3. We answer both questions negatively. We also refute a conjecture they made, resolve one further problem they posed, and make progress on another.

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).