Equitable Edge Coloring Research Articles

Equitable Edge Coloring of Splitting Graph of Some Classes of Wheel Graphs

The coloring of all the edges of a graph G with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of W n , D W n and G n by determining its edge chromatic number.

Equitable Edge Coloring of Splitting Graph of Some Classes of Wheel Graphs

The coloring of all the edges of a graph \(G\) with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of \(W_n\), \(DW_n\) and \(G_n\) by determining its edge chromatic number.

A STUDY ON FUZZY EQUITABLE EDGE COLORING OF WHEEL RELATED GRAPHS

Equitable edge coloring is a kind of graph labeling with the following restrictions. No two adjacent edges receive same label (color). and number of edges in any two color classes differ by at most one. In this work we are going to present the Fuzzy equitable edge coloring of some wheel related graphs.

FUZZY EQUITABLE EDGE COLORING OF SOME SIMPLE GRAPHS

In our earlier works, we have discussed about the equitable edge coloring of various classes of some simple graphs (or crisp graphs). In this Paper we are going to state and discuss the Fuzzy equitable edge coloring of some classes of simple graphs.

An equitable edge coloring of some classes of product of graphs

An equitable edge coloring of some classes of product of graphs

A Structure of 1-Planar Graph and Its Applications to Coloring Problems

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the $(p,1)$-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the $(p,1)$-total labelling number of every 1-planar graph $G$ is at most $\Delta(G)+2p-2$ provided that $\Delta(G)\geq 8p+2$ and $p\geq 2$, and show that every 1-planar graph has an equitable edge coloring with $k$ colors for any integer $k\geq 18$. These three results respectively generalize the main theorems of three different previously published papers.

On the Equitable Edge-Coloring of 1-Planar Graphs and Planar Graphs

An edge-coloring of a graph G is equitable if, for each vertex v of G, the number of edges of any one color incident with v differs from the number of edges of any other color incident with v by at most one. In the paper, we prove that every 1-planar graph has an equitable edge-coloring with k colors for any integer \(k\ge 21\), and every planar graph has an equitable edge-coloring with k colors for any integer \(k\ge 12\).

On the Adjacent Strong Equitable Edge Coloring ofPn∨Pn,Pn∨CnandCn∨Cn

A proper edge coloring of graph G is called equitable adjacent strong edge coloring if colored sets from every two adjacent vertices incident edge are different,and the number of edges in any two color classes differ by at most one,which the required minimum number of colors is called the adjacent strong equitable edge chromatic number. In this paper, we discuss the adjacent strong equitable edge coloring of join-graphs about P n ∨ P n , P n ∨ C n and C n ∨ C n .

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V,E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u) ≠ C(v) for ∨uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2, … k, we have ‖Ei| − |Ej‖ ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χ′ave(G) =min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number ofG. In this paper, we obtain the χ′ave(G) of some special graphs and present a conjecture.

The method of coloring in graphs and its application

Graph coloring has interesting real life applications in optimization and network design. In this paper some new results on the acyclic-edge coloring, f-edge coloring, g-edge cover coloring, (g, f)-coloring and equitable edge-coloring of graphs are introduced. In particular, some new results related to the above colorings obtained by the authors are given. Some new problems and conjectures are presented.

A Fast Algorithm for Computing a Nearly Equitable Edge Coloring with Balanced Conditions

We discuss the nearly equitable edge coloring problem on a multigraph and propose an efficient algorithm for solving the problem, which has a better time complexity than the previous algorithms. The coloring computed by our algorithm satisfies additional balanced conditions on the number of edges used in each color class, where conditions are imposed on the balance among all edges in the multigraph as well as the balance among parallel edges between each vertex pair. None of the previous algorithms are guaranteed to satisfy these balanced conditions simultaneously.

An Efficient Algorithm for the Nearly Equitable Edge Coloring Problem

An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that the problem of finding a nearly equitable edge coloring can be solved in O(m/k) time, where m and k are the numbers of edges and given colors, respectively. In this paper, we present a recursive algorithm that runs in O (mn log (m/(kn) + 1)) time, where n is the number of vertices. This algorithm improves the bestknown worst-case time complexity. When k = O(1), the time complexity of all known algorithms is O(m), which implies that this time complexity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any multigraph. Our result is the first that improves this time complexity when m/n grows to infinity; e.g., m = n for an arbitrary constant θ > 1. Submitted: July 2008 Reviewed: September 2008 Revised: October 2008 Accepted: October 2008 Final: November 2008 Published: December 2008 Article type: Regular paper Communicated by: D. Wagner This research was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan. A preliminary version of this paper appeared in Proceedings of ISAAC [10]. E-mail addresses: sharryx@al.cm.is.nagoya-u.ac.jp (Xuzhen Xie) yagiura@nagoya-u.jp (Mutsunori Yagiura) takao@al.cm.is.nagoya-u.ac.jp (Takao Ono) hirata@is.nagoya-u.ac.jp (Tomio Hirata) zwick@tau.ac.il (Uri Zwick) 384 X. Xie et al. An Efficient Algorithm for Nearly Equitable Edge Coloring