ON INCIDENCE COLORING OF SIGNED GRAPHS

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

Let G = (V, E) be a graph with vertex set V and edge set E. Given non negative integers r, s and t, an [r, s, t]-coloring of a graph G is a proper total coloring where the neighboring elements of G (vertices and edges) receive colors with a certain difference r between colors of adjacent vertices, a difference s between colors of adjacent edges and a difference t between colors of a vertex and an incident edge. Thus [r, s, t]-colorings generalize the classical colorings of graphs and can have applications in different fields like scheduling, channel assignment problem, etc. The [r, s, t]-chromatic number ? r,s,t (G) of G is the minimum k such that G admits an [r, s, t]-coloring. In our paper we propose several bounds for the [r, s, t]-chromatic number of the cartesian and direct products of some graphs.

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs (G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi |i = 1,2,⋯m}∪{vij |i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi ). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs (G ∘ H) and the exact values of χs (G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs (G ∘ H) ≤ χs (G) + χs (H); χs (Pm ∘ H) = χs (H) + 2; χs (K 1,m ∘ H) = χs (H) + 2; χs (Cn ∘ H) = χs (H) + 2, where n ≠ 5.

Indicated coloring of a graph G is a coloring in which there are two players Ann and Ben, Ann picks a vertex and Ben chooses a color for this vertex. The aim of Ann is to achieve a proper coloring of the whole graph G, while Ben tries to block the same. The smallest number of colors required for Ann to win the game on a graph G is called the indicated chromatic number of G and is denoted by \(\chi _i(G).\) In this paper, we prove that \(T\Box C_n, T\Box K_{n_1,n_2,\dots ,n_m}\) and \(K_{n_1,n_2,\dots ,n_m}\Box C_m\) are k-indicated colorable for all k greater than or equal to the indicated chromatic number of their corresponding Cartesian product, where T is any tree. Also we prove that \(\chi _i(K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}) =\chi (K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}).\) Finally we have given non-trivial examples of graphs G and H for which \(\chi _i(G\Box H) >\chi (G\Box H).\)

We study a new variant of connected coloring of graphs based on the concept of strong edge coloring (every color class forms an induced matching). In particular, an edge-colored path is strongly proper if its color sequence does not contain identical terms within a distance of at most two. A strong proper connected coloring of G is the one in which every pair of vertices is joined by at least one strongly proper path. Let spc(G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)$$\\end{document} denote the least number of colors needed for such coloring of a graph G. We prove that the upper bound spc(G)≤5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)\\le 5$$\\end{document} holds for any 2-connected graph G. On the other hand, we demonstrate that there are 2-connected graphs with arbitrarily large girth satisfying spc(G)≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)\\ge 4$$\\end{document}. Additionally, we prove that graphs whose cycle lengths are divisible by 3 satisfy spc(G)≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{spc}\\,}}(G)\\le 3$$\\end{document}. We also consider briefly other connected colorings defined by various restrictions on color sequences of connecting paths. For instance, in a nonrepetitive connected coloring of G, every pair of vertices should be joined by a path whose color sequence is nonrepetitive, that is, it does not contain two adjacent identical blocks. We demonstrate that 2-connected graphs are 15-colorable, while 4-connected graphs are 6-colorable, in the connected nonrepetitive sense. A similar conclusion with a finite upper bound on the number of colors holds for a much wider variety of connected colorings corresponding to fairly general properties of sequences. We end the paper with some open problems of concrete and general nature.

The concept of neutrosophic sets makes it easier to analyse values that are ambiguous or indeterminate. We illustrate the concept of single-valued neutrosophic R-dynamic vertex coloring in this study. In addition, the single valued neutrosophic dynamic vertex coloring of the Cartesian product of path with cycle, cycle with cycle, complete SVNG with complete SVNG and cycle has been determined. Also, we have discussed the single valued neutrosophic dynamic vertex coloring of join of complete SVNG with complete SVNG and cycle, cycle with cycle.

Small cell densification is one of the effective ideas addressing the demand for higher capacity in cellular networks. The major problem in such networks is the limited capacity of wireless backhaul links. Caching the popular files in the memories of small cell base stations (SBSs) is an effective solution to this problem. One of the main challenges in caching is choosing the files to be stored in the memory of SBSs, especially when some SBSs have overlapped coverage. In this article, we propose a new caching scheme by exploiting the concept of graph coloring. To this end, we model the network scenario by a graph. This graph is divided into four subgraphs, i.e., placement, access, SBS, and delivery graphs. Through some modifications to the SBS graph, we convert it to a graph that can be colored. Then, we propose a graph coloring-based scheme to cache the popular files. We also derive a lower-bound for the hit rate improvement of our scheme compared to the conventional scheme which caches the most popular files anywhere. We evaluate the performance of our proposed scheme through simulations. Our results show that by employing the proposed scheme in a typical network, the load on the backhaul links is reduced significantly in average compared to the conventional scheme and some previously proposed schemes.

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 proper coloring of a graph is a labeled partition of its vertices into parts which are independent sets. In this paper, given a positive integer j and a family ℱ of connected graphs, we consider proper colorings in which we require that the union of any j color classes induces a subgraph which has no copy of any member of ℱ. This generalizes some well-known types of proper colorings like acyclic colorings (where j = 2 and ℱis the collection of all even cycles) and star colorings (where j = 2 and ℱconsists of just a path on 4 vertices), etc. For this type of coloring, we obtain an upper bound of O(d(k − 1)/(k − j)) on the minimum number of colors sufficient to obtain such a coloring. Here, d refers to the maximum degree of the graph and k is the size of the smallest member of ℱ. For the case of j = 2, we also obtain lower bounds on the minimum number of colors needed in the worst case. As a corollary, we obtain bounds on the minimum number of colors sufficient to obtain proper colorings in which the union of any j color classes is a graph of bounded treewidth. In particular, using O(d8/7) colors, one can obtain a proper coloring of the vertices of a graph so that the union of any two color classes has treewidth at most 2. We also show that this bound is tight within a multiplicative factor of O((logd)1/7). We also consider generalizations where we require simultaneously for several pairs (ji, ℱi) (i = 1, …, l) that the union of any ji color classes has no copy of any member of ℱi and obtain upper bounds on the corresponding chromatic numbers. © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 213–234, 2011 © 2011 Wiley Periodicals, Inc.

Randomized online graph coloring

On vertex rankings of graphs and its relatives

The vertex coloring of graphs is a well-known coloring of graphs. In this coloring, all of the vertices are assigned colors in such a way that no two adjacent vertices have the same color. We can call this type of coloring P2 coloring, where P2 is a path graph. However, there are situations in which this type of coloring cannot give us the solution to the problem at hand. To answer such questions, in this article, we introduce a novel graph coloring called P3 coloring. A graph is called P3-colorable if we can assign colors to the vertices of the graph such that the vertices of every P3 path are distinct. The minimum number of colors required for a graph to have P3 coloring is called the P3 chromatic number. The aim of this article is, in general, to prove some basic results concerning this coloring, and, in particular, to compute the P3 chromatic number for different symmetric families of graphs.

formula omitted]-facial edge colorings of graphs

The edge weighting problem of graphs has interesting real life applications in the optimization and the network design, such as the file transfers in computer networks. A k-edge weighting of a graph G is a mapping w : E(G)®{1,, k}. An edge-weighting w induces a vertex coloring f (v) w given by I = e v f (v) w(e) w for each v IV(G) . An edge weighting w is vertex coloring if ( ) ( ) f u f v w w ¹ for each edge uvI E(G) . The current paper studies the parameter (G) e c , which is the minimum k for which G has a vertex coloring k-edge weighting. Exact value of (G) e c are determined for several double graph and join graph of some graphs.

Graph coloring is an old idea, bringing to mind such celebrated results as the four-color theorem and NP-hardness of the chromatic number of a graph, but not, for most people, the efficient approximation of derivatives. Yet during recent years graph coloring has emerged as a highly effective tool for approximating derivatives, whether Jacobians (matrices of first derivatives of a vector-valued function) or Hessians (matrices of second derivatives of a scalar-valued function). The main issue is how best to exploit the sparsity and symmetry in these matrices to compute them with the least work. Remarkably, the techniques for exploiting sparsity are essentially the same whether the derivatives are approximated by the very old technique of finite differences or computed exactly by the comparatively recent method of automatic differentiation. In both cases, the columns or rows (or both) of a derivative matrix must be partitioned into a small number of groups in order to reduce the number of function evaluations required. It is in formulating, analyzing, and designing algorithms for such problems that graph coloring has proved to be a powerful tool. Indeed, modern software for computing large, sparse Jacobians and Hessians relies on graph coloring algorithms to make the computations feasible. In "What Color Is Your Jacobian?~Graph Coloring for Computing Derivatives," Gebremedhin, Manne, and Pothen give a comprehensive survey of the work in this area during the past 25 years, presenting a unifying framework, several new algorithms, substantial computational results, and a survey of relevant theoretical results.