The Erdős-Gyárfás problem on generalized Ramsey numbers

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.

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

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

A proper edge coloring of a graph is an edge coloring such that no two adjacent edges are assigned the same color. The minimum number of colors needed to color edges of G is called the chromatic index χ/(G) of G. We introduce the new term Ce - V table for finding the inequality between and the sum of the terms domination number and the chromatic number of the induced sub graph. Here we introduced the applications of the edge coloring with the chromatic number in different dominations.

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

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.

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.

A variety of extremal problems are considered for the members of two particular families of simple 2-complexes. We let Bk denote the complex consisting of k 2-cells all sharing one single edge in common and Wk the complex consisting of k 2-cells sharing exactly one vertex in common. The problems treated here include the following questions for the Bk and the corresponding questions for the Wk.Suppose that for each 2-coloring of the 2-cells of a complex G, either one color contains a monochromatic copy of Bk or the other contains a monochromatic copy of Bℓ. How many 2-cells must G have? How many vertices? How many vertices and how many 2-cells must G have if one of Bk and Bℓ must be "faithfully imbedded" in one of the colors? How do these numbers compare with the "generalized Ramsey numbers" for the pairs (Bk, Bℓ)? What is the smallest integer f(n, k) such that each 2-complex on n vertices having f(n, k) 2-cells contains a copy of Bk?

Minimal coloring and strength of graphs

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.

On the complexity of rainbow coloring problems

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.

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′aa(G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′aa(G) ≤ 32Δ.

Colouring edges with many colours in cycles