Minimum Number Of Colors Research Articles

The 3-rainbow index of some graphs that constructed by joining a graph with a trivial graph

Let G = (V, E) be a nontrivial, finite, and connected graph. A tree T in an edge-colored graph is said to be rainbow tree, if no two edges on the tree have the same color. An edge-coloring of G is called 3-rainbow coloring, if for any three vertices in G there exists a rainbow tree connecting them. The 3-rainbow index of G, denoted by rx3(G), is the minimum number of colors that are needed in a 3-rainbow coloring of G. The join of G1 and G2, denoted by G = G1 + G2, is the graph with vertex set V(G1) ∪ V(G2) and edge set E(G1) ∪ E(G2) ∪ {uv|u ∊ V(G1),v ∊ V(G2)}. In this paper, we determine rx3(G) where G is the join of a graph with a trivial graph.

The 3-rainbow index of amalgamation of some graphs with diameter 2

Let G = (V, E) be a nontrivial, connected, and edge-colored graph with n vertices, and let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree, if no two edges of T receive the same color. A k-rainbow coloring of G is an edge coloring of G having property that for every set S of k vertices of G, there exists a rainbow tree T such that S ⊆ V (T). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rxk(G). The distance d(x,y) of two vertices x and y in G is the length of a shortest x — y path in G. The greatest distance between any two vertices in G is the diameter of G, denoted by diam(G). Let {G1, G2,…, Gt} be a finite collection of graphs and each graph Gi have a fixed vertex v0i called a terminal. The amalgamation of G1, G2,…, Gt, denoted by Amal(G1, v0l, G2, v02 …, Gt, v0t), is a graph obtained by taking all the s and identifying their terminals. In case Gi ≅ G and v0i = u, the amalgamation of G1, G2,…, Gt is denoted by Amal(G,t,u). In this paper, we determine the 3-rainbow index of amalgamation of some graphs Amal(G,t,u) with diam(G) = 2.

Can the hybrid colouring algorithm take advantage of multi-core architectures

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.

An upper bound for the choice number of star edge coloring of graphs

The star chromatic index of a multigraph G, denoted χs′(G), is the minimum number of colors needed to properly color the edges of G such that no path or cycle of length four is bi-colored. A multigraph G is star k-edge-colorable if χs′(G)≤k. Dvořák et al. (2013) proved that every subcubic multigraph is star 7-edge-colorable. They conjectured in the same paper that every subcubic multigraph should be star 6-edge-colorable. In this paper, we consider this problem in a more general setting, we investigate star list edge coloring of general graph G and obtain an upper bound for the choice number of star edge coloring of graphs, namely, we proved that χsl′≤⌈2Δ32(1Δ+2)12+2Δ⌉.

A Note on Strong Edge Coloring of Sparse Graphs

A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ′s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2k. Yang and Zhu [J. Graph Theory, 83, 334–339 (2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4kΔ − 2k in the square of the line graph of G, implying that χ′s(G) ≤ 4kΔ − 2k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most (4k − 1)Δ − 2k in the square of the line graph of G, implying that χ′s(G) ≤ (4k − 1)Δ − 2k + 1.

On the complexity of [formula omitted]-rainbow cycle colouring problems

An edge-coloured cycle is rainbow if all edges of the cycle have distinct colours. For k≥1, let Fk denote the family of all graphs with the property that any k vertices lie on a cycle. For G∈Fk, a k-rainbowcyclecolouring of G is an edge-colouring such that any k vertices of G lie on a rainbow cycle in G. The k-rainbowcycleindex of G, denoted by crxk(G), is the minimum number of colours needed in a k-rainbow cycle colouring of G. In this paper, we restrict our attention to the computational aspects of k-rainbow cycle colouring. First, we prove that the problem of deciding whether crx1=3 can be solved in polynomial time, but that of deciding whether crx1≤ℓ is NP-Complete, where ℓ≥4. Then we show that the problem of deciding whether crx2=3 can be solved in polynomial time, but those of deciding whether crx2≤4 or 5 are NP-Complete. Furthermore, we also consider the cases of crx3=3 and crx3≤4. Finally, we prove that the problem of deciding whether a given edge-colouring (with an unbounded number of colours) of a graph is a k-rainbow cycle colouring, is NP-Complete for k=1, 2 and 3, respectively. Some open problems for further study are mentioned.

Total Coloring Conjecture for Certain Classes of Graphs

A total coloring of a graph G is an assignment of colors to the elements of the graph G such that no two adjacent or incident elements 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 total coloring conjecture for certain classes of graphs of deleted lexicographic product, line graph and double graph.

A Study on Harmonious Coloring of Circulant Networks

Given a simple graph , a harmonious coloring of is the proper vertex coloring such that each pair of colors seems to appears together on at most one edge. The harmonious chromatic number of , denoted by is the minimal number of colors in a harmonious coloring of . In this paper we have determined the harmonious chromatic number of some classes of Circulant Networks.

The chromatic number of the space

The chromatic number of the space with the metric with forbidden distances is studied. It is defined as the minimum number of colours needed to colour all points in the space in such a way that no two points lying at a forbidden distance from each other in the metric have the same colour. The asymptotic growth of the chromatic numbers as is estimated. The instrument of study is the linear algebra method, which reduces the problem of estimating chromatic numbers to another convex extremum problem. A numerical solution of this problem makes it possible to derive sharp estimates for the constants present in the bases of the lower asymptotic estimates for the chromatic numbers of multidimensional real spaces with several forbidden distances. The estimates obtained are optimal within the framework of the method proposed. Bibliography: 27 titles.

On dominated coloring of graphs and some Nordhaus–Gaddum-type relations

The dominated coloring of a graph $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$, denoted by $\chi_{dom}(G)$. In this paper, dominated coloring of graphs is compared with (open) packing number of $G$ and it is shown that if $G$ is a graph of order $n$ with $diam(G)\geq3$, then $\chi_{dom}(G)\leq n-\rho(G)$ and if $\rho_0 (G)=2n/3$, then $\chi_{dom}(G)= \rho_0 (G)$, and if $\rho(G)=n/2$, then $\chi_{dom}(G)=\rho(G)$. The dominated chromatic numbers of the corona of two graphs are investigated and it is shown that if $\mu(G)$ is the Mycielsky graph of $G$, then we have $\chi_{dom} (\mu(G))=\chi_{dom} (G)+1$. It is also proved that the Vizing-type conjecture holds for dominated colorings of the direct product of two graphs. Finally we obtain some Nordhaus--Gaddum-type results for the dominated chromatic number $\chi_{dom}(G)$.

More on the total dominator chromatic number of a graph

Let G be a simple graph. A total dominator coloring of G, is a proper coloring of the vertices of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number of G, is the minimum number of colors among all total dominator coloring of G. The neighbourhood corona of two graphs G1 and G2 is denoted by G1 ⋆ G2 and is the graph obtained by taking one copy of G1 and |V(G1)| copies of G2, and joining the neighbours of the ith vertex of G1 to every vertex in the ith copy of G2. In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of r-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the TDC-number of G. We study the stability and bondage number of certatin graphs.

On the Rainbow and Strong Rainbow Coloring of Comb Product Graphs

Let G=(V,E) be a simple, nontrivial, finite, connected and undirected graph. Let c be a coloring c: E(G)g{1,2,…,k}, kdN. A path of the edges of a colored graph is said to be a rainbow path if no two edges on the path have the same color but the adjacent edges may be colored by the same colors. An edge colored graph G is rainbow connected if there exists a rainbow u-v path for every two vertices u and v of G. Furthermore, for any two vertices u and v of G, a rainbow u-v geodesic in G is a rainbow u-v path of length d(u,v), where d(u,v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u-v geodesic for any two vertices u and v in G. The rainbow and strong rainbow connection numbers of a graph G, denoted by rc(G) and src(G) respectively, are the minimum number of colors that are needed in order to make G rainbow and strongly rainbow connected, respectively. Some results have shown the lower and upper bound of rc(G) and src(G), but most of them are not sharp. Thus, finding an exact value of rc(G) and src(G) are significantly useful. In this paper, we study the exact values of rainbow and strong rainbow connection numbers of comb product graphs.

On conflict-free connection of graphs

An edge-colored graph G is conflict-free connected if, between each pair of distinct vertices of G, there exists a path in G containing a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we firstly determine all trees T of order n for which cfc(T)=n−t, where t≥1 and n≥2t+2. Secondly, we prove that let G be a graph of order n, then 1≤cfc(G)≤n−1, and characterize the graphs G with cfc(G)=1,n−4,n−3,n−2,n−1, respectively. Finally, we get the Nordhaus–Gaddum-type result for the conflict-free connection number of graphs, and prove that if G and G¯ are connected graphs of order n (n≥4), then 4≤cfc(G)+cfc(G¯)≤n and 4≤cfc(G)⋅cfc(G¯)≤2(n−2), moreover, cfc(G)+cfc(G¯)=n or cfc(G)⋅cfc(G¯)=2(n−2) if and only if one of G and G¯ is a tree with maximum degree n−2 or a path of order 5, and the lower bounds are sharp.

Strong Chromatic Index of Graphs With Maximum Degree Four

A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In 1985, Erdős and Nešetřil conjectured that every graph with maximum degree $\Delta$ has a strong edge-coloring using at most $\frac{5}{4}\Delta^2$ colors if $\Delta$ is even, and at most $\frac{5}{4}\Delta^2 - \frac{1}{2}\Delta + \frac{1}{4}$ if $\Delta$ is odd. Despite recent progress for large $\Delta$ by using an iterative probabilistic argument, the only nontrivial case of the conjecture that has been verified is when $\Delta = 3$, leaving the need for new approaches to verify the conjecture for any $\Delta\ge 4$. In this paper, we apply some ideas used in previous results to an upper bound of 21 for graphs with maximum degree 4, which improves a previous bound due to Cranston in 2006 and moves closer to the conjectured upper bound of 20.

Local antimagic vertex coloring of unicyclic graphs

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈<sub>E(u)</sub> f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

A Remark on Rainbow 6-Cycles in Hypercubes

We call an edge-coloring of a graph [Formula: see text] a rainbow coloring if the edges of [Formula: see text] are colored with distinct colors. For every even positive integer [Formula: see text], let [Formula: see text] denote the minimum number of colors required to color the edges of the [Formula: see text]-dimensional cube [Formula: see text], so that every copy of [Formula: see text] is rainbow. Faudree et al. [6] proved that [Formula: see text] for [Formula: see text] or [Formula: see text]. Mubayi et al. [8] showed that [Formula: see text]. In this note, we show that [Formula: see text]. Moreover, we obtain the number of 6-cycles of [Formula: see text].

Bounds on the Dynamic Chromatic Number of a Graph in Terms of its Chromatic Number

A vertex coloring of a graph is called dynamic if the neighborhood of any vertex of degree at least 2 contains at least two vertices of distinct colors. Similarly to the chromatic number χ(G) of a graph G, one can define its dynamic number χd(G) (the minimum number of colors in a dynamic coloring) and dynamic chromatic number χ2(G) (the minimum number of colors in a proper dynamic coloring). We prove that χ2(G) ≤ χ(G) · χd(G) and construct an infinite series of graphs for which this bound on χ2(G) is tight. For a graph G, set $$ k=\left\lceil \frac{2\Delta (G)}{\delta (G)}\right\rceil $$ We prove that χ2(G) ≤ (k+1)c. Moreover, in the case where k ≥ 3 and Δ(G) ≥ 3, we prove the stronger bound χ2(G) ≤ kc.

The $$(k,\ell )$$ ( k , ℓ ) -proper index of graphs

A tree T in an edge-colored graph is called a proper tree if no two adjacent edges of T receive the same color. Let G be a connected graph of order n and k be an integer with $$2\le k \le n$$ . For $$S\subseteq V(G)$$ and $$|S| \ge 2$$ , an S-tree is a tree containing the vertices of S in G. A set $$\{T_1,T_2,\ldots ,T_\ell \}$$ of S-trees is called internally disjoint if $$E(T_i)\cap E(T_j)=\emptyset $$ and $$V(T_i)\cap V(T_j)=S$$ for $$1\le i\ne j\le \ell $$ . For a set S of k vertices of G, the maximum number of internally disjoint S-trees in G is denoted by $$\kappa (S)$$ . The k-connectivity $$\kappa _k(G)$$ of G is defined by $$\kappa _k(G)=\min \{\kappa (S)\mid S$$ is a k-subset of $$V(G)\}$$ . For a connected graph G of order n and for two integers k and $$\ell $$ with $$2\le k\le n$$ and $$1\le \ell \le \kappa _k(G)$$ , the $$(k,\ell )$$ -proper index $$px_{k,\ell }(G)$$ of G is the minimum number of colors that are required in an edge-coloring of G such that for every k-subset S of V(G), there exist $$\ell $$ internally disjoint proper S-trees connecting them. In this paper, we show that for every pair of positive integers k and $$\ell $$ with $$k \ge 3$$ and $$\ell \le \kappa _k(K_{n,n})$$ , there exists a positive integer $$N_1=N_1(k,\ell )$$ such that $$px_{k,\ell }(K_n) = 2$$ for every integer $$n \ge N_1$$ , and there exists also a positive integer $$N_2=N_2(k,\ell )$$ such that $$px_{k,\ell }(K_{m,n}) = 2$$ for every integer $$n \ge N_2$$ and $$m=O(n^r) (r \ge 1)$$ . In addition, we show that for every $$p \ge c\root k \of {\frac{\log _a n}{n}}$$ ( $$c \ge 5$$ ), $$px_{k,\ell }(G_{n,p})\le 2$$ holds almost surely, where $$G_{n,p}$$ is the Erdős–Renyi random graph model.

Facial Rainbow Edge-Coloring of Plane Graphs

An edge-coloring of a loopless plane graph G is a facial rainbow edge-coloring if any two edges of G contained in the same facial path have distinct colors. The facial rainbow edge-number of a graph G, denoted $$\mathrm {erb}(G)$$ , is the minimum number of colors that are necessary in any facial rainbow edge-coloring. In the present note we prove that $$\mathrm {erb}(G) \le \lfloor \frac{3}{2} (L(G) + 1) \rfloor $$ for all connected loopless plane graphs, where L(G) is the length of the longest facial path of G. This bound is tight. For the family of all 3-connected plane graphs this bound is improved to $$L(G) + 2$$ . For trees there is $$\mathrm {erb}(G) \le \lfloor \frac{3}{2} L(G) \rfloor $$ which is also tight. Moreover, if G is a tree with $$L(G) \ge 7$$ and without degree two vertices, then $$\mathrm {erb}(G) = L(G)$$ .

2-Distance vertex-distinguishing index of subcubic graphs

A 2-distance vertex-distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets of colors. The 2-distance vertex-distinguishing index $$\chi ^{\prime }_{\mathrm{d2}}(G)$$ of G is the minimum number of colors needed for a 2-distance vertex-distinguishing edge coloring of G. Some network problems can be converted to the 2-distance vertex-distinguishing edge coloring of graphs. It is proved in this paper that if G is a subcubic graph, then $$\chi ^{\prime }_{\mathrm{d2}}(G)\le 6$$ . Since the Peterson graph P satisfies $$\chi ^{\prime }_{\mathrm{d2}}(P)=5$$ , our solution is within one color from optimal.