ON DYNAMIC COLORING OF WEB GRAPH

Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number chi _d(G) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.

We investigate the generalized line graph, total graph and middle graph of the given graphs. First, replace a given graph G with a multigraph M having the same point set as G, then it constructs the total graph T(G) from G. Two suitable subgraphs Mx(G), Lx(G), of Tx(G) are considered as extensions of middle graph M(G) and line graph L(G) respectively.We also consider the connectivities of graphs which are obtained as a special case of Lx(G), Mx(G) and Tx(G). Then we get some extensions of established theorems.

Irwing and Manlove instigate the knowledge of b -colorings and provided some inequalities for extreme bounds to explore the parameter on b -coloring. He denoted m-degree m(G) to determine the b -chromatic number. Here the parameter on b-coloring of Line graph of Tadpole Graph, Middle graph and Total graph of tadpole graph are being investigated. Also comparison of parameters on coloring and b-coloring has been done.

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

The two fastest-growing subfields of graph theory are graph coloringand topological indices. Graph coloring is assigning the colors/values to theedges/vertices or both. A proper coloring of the graph G is assigning colors/valuesto the vertices/edges or both so that no two adjacent vertices/edges share the samecolor/value. Recently, studies involving Chromatic Topological indices that dealtwith different graph coloring were studied. In such studies, the vertex degrees getreplaced with the colors, and the computation is carried out based on the topologicalindex of our choice. We focus on b-Chromatic Zagreb indices and b-Chromaticirregularity indices in this work. This paper discusses the b-Chromatic Zagreb indicesand b-Chromatic irregularity indices of the gear graph, star graph, and itsderived graphs such as the line and middle graph.

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.

Graph labeling is the assignment of integers to edges or vertices or both depending on some factors which is consequential in the field of graph theory. In this paper, anti-magic graph labeling of two graphs, middle graph and total graph of path graph have been mainly focused. Modified proofs of theorems on anti-magic graph labeling have been introduced here. Both of the theorems are applicable for strong anti-magic graph. The correctness of the proofs has been demonstrated through various examples.

In this article, a new idea of fuzzy fractional coloring of fuzzy graph is presented and fuzzy fractional chromatic number is defined. A relationship between fuzzy fractional chromatic number and fuzzy fractional clique number is established. Some properties of fuzzy chromatic number of fuzzy graphs and fuzzy fractional chromatic number of fuzzy graphs are proved and the concept of k-strong adjacent vertices is introduced. Fuzzy chromatic number and fuzzy fractional chromatic number have been calculated on lexicographic product of two fuzzy graphs. Also, fuzzy chromatic number, independence number and fuzzy fractional chromatic number have been investigated on disjoint union of two fuzzy graphs. Lastly, a real life application of fuzzy fractional coloring on fuzzy graph is discussed.

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme.

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.

For a simple connected graph G, let c : V(G) → ℕ be a vertex coloring of G, where adjacent vertices may be colored the same. The neighborhood color set of a vertex v, denoted by NC(v), is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) ≠ NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χs(G). In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

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

Relations of vertex distinguishing total chromatic numbers between a subgraph and its supergraph

A vertex coloring of a graph G is a mapping that allots colors to the vertices of G. Such a coloring is said to be a proper vertex coloring if two vertices joined by an edge receive different colors. The chromatic number χ ( G ) is the least number of colors used in a proper vertex coloring. In this paper, we compute the χ of certain distance graphs whose distance set elements are (a) a finite set of Catalan numbers, (b) a finite set of generalized Catalan numbers, (c) a finite set of Hankel transform of a transformed sequence of Catalan numbers. Then while discussing the importance of minimizing interference in wireless networks, we probe how a vertex coloring problem is related to minimizing vertex collisions and signal clashes of the associated interference graph. Then when investigating the χ of certain G ( V , D ) and graphs with interference, we also compute certain lower and upper bound for χ of any given simple graph in terms of the average degree and Laplacian operator. Besides obtaining some interesting results we also raised some open problems.