Maximum Number Of Colors Research Articles

On achromatic number of central graph of some graphs

The concept of coloring a graph will lead to the definition of a complete n- coloring of a graph G which results the achromatic number ψ(G) where the maximum number of colors required for the points of G in which every pair of colors appears on at least one pair of adjacent vertices. In this paper, we obtain the achromatic number for the Central graph of Ladder graph, Central graph of Dutch-Windmill graph, Central graph of Fan graph and Central graph of Flower graph is denoted as ψ[C(L_{n})], ψ[C(D₃⁽ⁿ⁾)], ψ[C(F_{m,n})] and ψ[C(FL_{n})] respectively.

Anti-Ramsey Problems in Complete Bipartite Graphs for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles and Matchings

A subgraph H of an edge-colored graph G is rainbow if all of its edges have different colors. The anti-Ramsey number is the maximum number of colors in an edge-coloring of G with no rainbow copy of H. Originally a complete graph was considered as G. In this paper, we consider a complete bipartite graph as the host graph and discuss some results for the graph H being hamiltonian cycle and perfect matching. Let $$c(K_{p,p},t)$$ and $$m(K_{p,p},t)$$ be the maximum number of colors in an edge-coloring of the complete bipartite graph $$K_{p,p}$$ not having t edge-disjoint rainbow hamiltonian cycles and perfect matchings, respectively. We prove that $$c(K_{p,p},t)=p^2-p+t$$ for $$t \ge 2$$ , $$p\ge 4t-1$$ and $$m(K_{p,p},t)=p^2-2p+t+1$$ for $$t \ge $$ 2, $$p\ge 2t+8$$ .

Planar anti-Ramsey numbers of matchings

Given a positive integer n and a planar graph H, let Tn(H) be the family of all plane triangulations T on n vertices such that T contains a subgraph isomorphic to H. The planar anti-Ramsey number ofH, denoted arP(n,H), is the maximum number of colors in an edge-coloring of a plane triangulation T∈Tn(H) such that T contains no rainbow copy of H. In this paper we study planar anti-Ramsey numbers of matchings. For all t≥1, let Mt denote a matching of size t. We prove that for all t≥6 and n≥3t−6, 2n+3t−15≤arP(n,Mt)≤2n+4t−14, which significantly improves the existing lower and upper bounds for arP(n,Mt). It seems that for each t≥6, the lower bound we obtained is the exact value of arP(n,Mt) for sufficiently large n. This is indeed the case for M6. We prove that arP(n,M6)=2n+3 for all n≥30.

On chromatic indices of finite affine spaces

The pseudoachromatic index of the finite affine space $\mathrm{AG}(n,q),$ denoted by $\psi'(\mathrm{AG}(n,q)),$ is the the maximum number of colors in any complete line-coloring of $\mathrm{AG}(n,q).$ When the coloring is also proper, the maximum number of colors is called the achromatic index of $\mathrm{AG}(n,q).$ We prove that if $n$ is even then $\psi'(\mathrm{AG}(n,q))\sim q^{1.5n-1}$; while when $n$ is odd the value is bounded by $q^{1.5(n-1)}<\psi'(\mathrm{AG}(n,q))<q^{1.5n-1}$. Moreover, we prove that the achromatic index of $\mathrm{AG}(n,q)$ is $q^{1.5n-1}$ for even $n,$ and we provides the exact values of both indices in the planar case.

Maximum colorful independent sets in vertex-colored graphs

In this paper we study the problem of finding a maximum colorful independent set in vertex-colored graphs. Specifically, given a graph with colored vertices, we wish to find an independent set containing the maximum number of colors. Here we aim to give a dichotomy overview on the complexity of this problem. We first show that the problem is NP-hard even for some cases where the maximum independent set problem is easy, such as cographs and P5-free graphs. Next, we provide polynomial-time algorithms for cluster graphs and trees.

Some extremal results on the colorful monochromatic vertex-connectivity of a graph

A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph $G$, the \emph{monochromatic vertex-connection number}, denoted by $mvc(G)$, is defined to be the maximum number of colors used in an \emph{MVC-coloring} of $G$. These concepts of vertex-version are natural generalizations of the colorful monochromatic connectivity of edge-version, introduced by Caro and Yuster. In this paper, we mainly investigate the Erd\H{o}s-Gallai-type problems for the monochromatic vertex-connection number $mvc(G)$ and completely determine the exact value. Moreover, the Nordhaus-Gaddum-type inequality for $mvc(G)$ is also given.

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring, the problem of determining whether a given graph has Grundy number at least k. We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Coloring (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected).We show that Grundy Coloring can be solved in time O∗(2.443n) and Weak Grundy Coloring in time O∗(2.716n) on graphs of order n. While Grundy Coloring and Weak Grundy Coloring are known to be solvable in time O∗(2O(wk)) for graphs of treewidth w (where k is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O∗(2o(wlogw)). We also describe an O∗(22O(k)) algorithm for Weak Grundy Coloring, which is therefore FPT for the parameter k. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for Grundy Coloring). Although we do not know whether Grundy Coloring is in FPT, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for Grundy Coloring and Weak Grundy Coloring on apex-minor graphs. In stark contrast with the two other problems, we show that Connected Grundy Coloring is NP-complete already for k=7 colors.

Rainbow matchings in edge-colored complete split graphs

In 1973, Erdős et al. introduced the anti-Ramsey number for a graph G in Kn, which is defined to be the maximum number of colors in an edge-coloring of Kn which does not contain any rainbow G. This is always regarded as one of rainbow generalizations of the classic Ramsey theory. Since then the anti-Ramsey numbers for several special graph classes in complete graphs have been determined. Also, the researchers generalized the host graph for the anti-Ramsey number from the complete graph to general graphs, including bipartite graphs, complete split graphs, planar graphs, and so on. In this paper, we study the anti-Ramsey number of matchings in the complete split graph. Since the complete split graph contains the complete graph as a subclass, the results in this paper cover the previous results about the anti-Ramsey number of matchings in the complete graph.

On N2-vertex coloring of graphs

Let [Formula: see text] be a graph. A vertex coloring [Formula: see text] of [Formula: see text] is called [Formula: see text]-vertex coloring if [Formula: see text] for every vertex [Formula: see text] of [Formula: see text], where [Formula: see text] is the set of colors of vertices adjacent to [Formula: see text]. Let [Formula: see text] be the maximum number of colors used in an [Formula: see text]-vertex coloring of [Formula: see text]. We provide some lower and upper bounds for [Formula: see text] in terms of girth, diameter or size of [Formula: see text]. Also, for every tree [Formula: see text], we obtain [Formula: see text].

On the Pseudoachromatic Index of the Complete Graph III

Let $ \Pi_q $ be the projective plane of order $ q $, let $\psi(m):=\psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $ m $, let $ a\in \{ 3,4,\dots,\tfrac{q}{2}+1 \} $ and $ m_a=(q+1)^2-a $. In this paper, we improve the upper bound of $ \psi(m) $ given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89--97] and Jamison [Discrete Math. 74 (1989), 99--115] in the following values: if $ x\geq 2 $ is an integer and $m\in \{4x^2-x,\dots,4x^2+3x-3\}$ then $\psi(m) \leq 2x(m-x-1)$. On the other hand, if $ q $ is even and there exists $ \Pi_q $ we give a complete edge-colouring of $ K_{m_a} $ with $(m_a-a)q$ colours. Moreover, using this colouring we extend the previous results for $a=\{-1,0,1,2\}$ given by Araujo-Pardo et al. in [J Graph Theory 66 (2011), 89--97] and [Bol. Soc. Mat. Mex. (2014) 20:17--28] proving that $\psi(m_a)=(m_a-a)q$ for $ a\in \{3,4,\dots,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} $.

Polychromatic colorings on the hypercube

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.

Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

An intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cyclic intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. Denote by w(G) (wc(G)) and W(G) (Wc(G)) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph G, respectively. We present some new sharp bounds on w(G) and W(G) for multigraphs G satisfying various conditions. In particular, we show that if G is a 2-connected multigraph with an interval coloring, then W(G)≤1+|V(G)|2(Δ(G)−1). We also give several results towards the general conjecture that Wc(G)≤|V(G)| for any triangle-free graph G with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most 4.

Thoroughly dispersed colorings

AbstractWe consider (not necessarily proper) colorings of the vertices of a graph where every color is thoroughly dispersed, that is, appears in every open neighborhood. Equivalently, every color is a total dominating set. We define as the maximum number of colors in such a coloring and as the fractional version thereof. In particular, we show that every claw‐free graph with minimum degree at least two has and this is best possible. For planar graphs, we show that every triangular disc has and this is best possible, and that every planar graph has and this is best possible, while we conjecture that every planar triangulation has . Further, although there are arbitrarily large examples of connected, cubic graphs with , we show that for a connected cubic graph . We also consider the related concepts in hypergraphs.

Anti-Ramsey numbers for matchings in regular bipartite graphs

Let [Formula: see text] be a family of graphs. The anti-Ramsey number [Formula: see text] for [Formula: see text] in the graph [Formula: see text] is the maximum number of colors in an edge coloring of [Formula: see text] that does not have any rainbow copy of any graph in [Formula: see text]. In this paper, we consider the anti-Ramsey number for matchings in regular bipartite graphs and determine its value under several conditions.

The minimum feedback arc set problem and the acyclic disconnection for graphs

A minimum feedback arc set of a digraph D is a minimum set of arcs which removal leaves the resultant graph free of directed cycles; its cardinality is denoted by τ1(D). The acyclic disconnection of D, ω⃗(D), is defined as the maximum number of colors in a vertex coloring of D such that every directed cycle of D contains at least one monochromatic arc. In this article we study the relationship between the minimum feedback arc set and the acyclic disconnection of a digraph, we prove that the acyclic disconnection problem is NP-complete. We define the acyclic disconnection and the minimum feedback for graphs. We also prove that ω⃗(G)+τ1(G)=|V(G)| if G is a wheel, a grid or an outerplanar graph.

Extremal coloring for the anti-Ramsey problem of matchings in complete graphs

Given a graph G, the anti-Ramsey number \(AR(K_n,G)\) is defined to be the maximum number of colors in an edge-coloring of \(K_n\) which does not contain any rainbow G (i.e., all the edges of G have distinct colors). The anti-Ramsey number was introduced by Erdős et al. (Infinite and finite sets, pp 657–665, 1973) and so far it has been determined for several special graph classes. Another related interesting problem posed by Erdős et al. is the uniqueness of the extremal coloring for the anti-Ramsey number. Contrary to the anti-Ramsey number, there are few results about the extremal coloring. In this paper, we show the uniqueness of such extremal coloring for the anti-Ramsey number of matchings in the complete graph.

Anti-Ramsey numbers for disjoint copies of graphs

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph \(G\) and a positive integer \(n\), the anti-Ramsey number \(ar(n,G)\) is the maximum number of colors in an edge-coloring of \(K_n\) with no rainbow copy of \(H\). Anti-Ramsey numbers were introduced by Erdȍs, Simonovits and Sos and studied in numerous papers. Let \(G\) be a graph with anti-Ramsey number \(ar(n,G)\). In this paper we show the lower bound for \(ar(n,pG)\), where \(pG\) denotes \(p\) vertex-disjoint copies of \(G\). Moreover, we prove that in some special cases this bound is sharp.

On improper interval edge colourings

We study improper interval edge colourings, defined by the requirement that the edge colours around each vertex form an integer interval. For the corresponding chromatic invariant (being the maximum number of colours in such a colouring), we present upper and lower bounds and discuss their qualities; also, we determine its values and estimates for graphs of various families, like wheels, prisms or complete graphs. The study of this parameter was inspired by the interval colouring, introduced by Asratian, Kamalian (1987). The difference is that we relax the requirement on the original colouring to be proper.

Total monochromatic connection of graphs

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A path in a total-colored graph is a total monochromatic path if all the edges and internal vertices on the path have the same color. A total-coloring of a graph is a total monochromatically-connecting coloring (TMC-coloring, for short) if any two vertices of the graph are connected by a total monochromatic path of the graph. For a connected graph G, the total monochromatic connection number, denoted by tmc(G), is defined as the maximum number of colors used in a TMC-coloring of G. These concepts are inspired by the concepts of monochromatic connection number mc(G), monochromatic vertex connection number mvc(G) and total rainbow connection number trc(G) of a connected graph G. Let l(T) denote the number of leaves of a tree T, and let l(G)=max{l(T)|T is a spanning tree ofG} for a connected graph G. In this paper, we show that there are many graphs G such that tmc(G)=m−n+2+l(G), and moreover, we prove that for almost all graphs G, tmc(G)=m−n+2+l(G) holds. Furthermore, we compare tmc(G) with mvc(G) and mc(G), respectively, and obtain that there exist graphs G such that tmc(G) is not less than mvc(G) and vice versa, and that tmc(G)=mc(G)+l(G) holds for almost all graphs. Finally, we prove that tmc(G)≤mc(G)+mvc(G), and the equality holds if and only if G is a complete graph.

Anti-Ramsey numbers for matchings in 3-regular bipartite graphs

The anti-Ramsey number AR(Kn, H) was introduced by Erdős, Simonovits and Sós in 1973, which is defined to be the maximum number of colors in an edge coloring of the complete graph Kn without any rainbow H. Later, the anti-Ramsey numbers for several special graph classes in complete are determined. Moreover, researchers generalized the host graph Kn to other graphs, in particular, to complete bipartite graphs and regular bipartite graphs. Li and Xu (2009) [18] proved that: Let G be a k-regular bipartite graph with n vertices in each partite set, then AR(G,mK2)=k(m−2)+1 for all m ≥ 2, k ≥ 3 and n>3(m−1). In this paper, we consider the anti-Ramsey number for matchings in 3-regular bipartite graphs. By using the known result that the vertex cover equals the size of maximum matching in bipartite graphs, we prove that AR(G,mK2)=3(m−2)+1 for n>32(m−1) when G is a 3-regular bipartite graph with n vertices in each partite set.