Minimal coloring and strength of graphs

A dominator coloring of a graph $G$ is a proper coloring in which every vertex of $G$ dominates at least one color class. The dominator chromatic number of $G, \chi_d(G)$, is defined by the minimum number of colors needed in a dominator coloring of $G$. A total dominator coloring of $G$ is a proper coloring of $G$ with the extra property that every vertex in $G$ properly dominates a color class. The total dominator chromatic number of $G, \chi_{t d}(G)$, is the minimum number of colors needed in a total dominator coloring of $G$. In this paper, we obtain dominator and total dominator chromatic number of Mongolian tent graphs and fire cracker graphs.

Colouring edges with many colours in cycles

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.

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figures

Coloring point adalah salah satu topik dalam Teori Graf, yaitu tentang mewarnai semua titik pada grafik, sehingga tidak ada dua titik tetangga yang memiliki warna yang sama. Salah satu aplikasi adalah sistem lampu lalu lintas, yang dapat membantu meningkatkan efektivitas lampu lalu lintas untuk mencegah dan mengatasi masalah kemacetan. Tujuan utama titik pewarnaan adalah menggunakan warna minimum yang berbeda untuk mewarnai semua titik pada grafik. Jumlah minimum warna yang digunakan disebut nomor Chromatic. Semakin sedikit warna yang digunakan, semakin efektif solusinya. Jumlah warna dalam sistem lampu lalu lintas menunjukkan jumlah kondisi untuk mengelola lampu lalu lintas. Ada banyak algoritme titik pewarnaan yang berbeda; tiga di antaranya adalah algoritma LDO, SDO dan IDO. Dalam tulisan ini, kami akan menerapkan dan membandingkan ketiga algoritma ini dengan grafik lampu lalu lintas dalam melintasi lima Kota Tua Ampenan. Kami memilih persimpangan jalan ini karena ini adalah salah satu persimpangan ramai di kota Mataram, terutama di pagi hari, jam kerja dan malam hari. Berdasarkan penelitian kami, untuk kasus ini algoritma LDO dan IDO lebih efektif daripada algoritma SDO

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ?14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.

Fix positive integers p and q with 2 ⩽ q ⩽ ( p 2 ) . An edge coloring of the complete graph K n is said to be a ( p , q ) -coloring if every K p receives at least q different colors. The function f ( n , p , q ) is the minimum number of colors that are needed for K n to have a ( p , q ) -coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that f ( n , p , p ) is polynomial in n and asked to determine the maximum q, depending on p, for which f ( n , p , q ) is subpolynomial in n. We prove that the answer is p − 1 .

On the chromatic index of multigraphs and a conjecture of Seymour (I)

This paper studies t‐interleaving on two‐dimensional tori. Interleaving has applications in distributed data storage and burst error correction, and is closely related to Lee metric codes. A t‐interleaving of a graph is defined as a vertex coloring in which any connected subgraph of t or fewer vertices has a distinct color at every vertex. We say that a torus can be perfectly t‐interleaved if its t‐interleaving number (the minimum number of colors needed for a t‐interleaving) meets the sphere‐packing lower bound, $\lceil t^2/2 \rceil$. We show that a torus is perfectly t‐interleavable if and only if its dimensions are both multiples of $\frac{t^2+1}{2}$ (if t is odd) or t (if t is even). The next natural question is how much bigger the t‐interleaving number is for those tori that are not perfectly t‐interleavable, and the most important contribution of this paper is to find an optimal interleaving for all sufficiently large tori, proving that when a torus is large enough in both dimensions, its t‐interleaving number is at most just one more than the sphere‐packing lower bound. We also obtain bounds on t‐interleaving numbers for the cases where one or both dimensions are not large, thus completing a general characterization of t‐interleaving numbers for two‐dimensional tori. Each of our upper bounds is accompanied by an efficient t‐interleaving scheme that constructively achieves the bound.

In the previous chapters, we have been considering vertex colourings. Now we will look at edge colourings of a graph. We will say that two edges are adjacent if they have a common vertex. We would like to colour the edges “properly” in the sense that no two adjacent edges receive the same colour. Given a graph X, we define the edge chromatic number denoted Xe (X) to be the minimum number of colours needed to properly colour the edges.

On the complexity of rainbow coloring problems

On the (di)graphs with (directed) proper connection number two

Graph colouring is a complex computational problem that focuses on colouring all vertices of a given graph using a minimum number of colours. However, adjacent vertices are restricted from receiving the same colour. Over recent decades, various algorithms have been proposed and implemented to solve such a problem. An interesting algorithm is the hybrid colouring algorithm, which was developed in 1999 by Galinier and Hao. The hybrid colouring algorithm was widely regarded at the time as one of the best performing algorithms for graph colouring. Nowadays, high-performance out-of-order multi-cores have emerged; consequently, executing applications faster and efficiently. Thus, the objective of this paper is to analyse whether the hybrid colouring algorithm can take advantage of multi-core architectures, in terms of performance, or not. For this purpose, we propose and implement a parallel version of the hybrid colouring algorithm that takes advantage of all hardware resources. Several experiments were performed on a machine with two Intel(R) Xeon(R) CPU E5-2630 processors, thus having a total of 24 cores. The experiment proved that the parallel hybrid colouring algorithm, using multi-core architectures, is a significant improvement over the original because it achieves enhancements of up to 40% in terms of the distance to the best chromatic number found in the literature. The expected contribution of this paper is to encourage developers to take advantage of high performance out-of-order multi-cores to solve complex computational problems.

Let 𝐺 be a connected graph and П={𝑆1,𝑆2,…,𝑆𝑘} be an ordered partition of 𝑉(𝐺). Let 𝑆𝑖 is a set of color classes using colors 1,2,...,𝑘 where 𝑘 as positive integer. The color code 𝑐П(𝑣) of vertex 𝑣 in 𝐺 with respect to П is defined as 𝑘-vector, 𝑐П(𝑣)=(𝑑(𝑣,𝑆1),𝑑(𝑣,𝑆2),…,𝑑(𝑣,𝑆𝑖)) where 𝑑(𝑣,𝑆𝑖)=𝑚𝑖𝑛{𝑑(𝑣,𝑥)|𝑥∈𝑆𝑖} for 1≤𝑖≤𝑘. If each of vertices in 𝐺 have distinct color codes, then 𝑐 is called as locating coloring of 𝐺. The minimum number of colors that are used for locating coloring is called as locating chromatic number of 𝐺, denoted by 𝜒𝐿(𝐺). In this article we will discuss the chromatic number of the side amalgamation location of the circle graph 𝑎𝑚𝑎𝑙𝑠(𝐶𝑛𝑗;𝑣𝑗,𝑙𝑣𝑗,𝑛) with 𝑛=3,4,1≤𝑗≤𝑚, and 𝑚≥2.

Colouring graphs with bounded generalized colouring number