Remarks on Extremal Overfull Graphs

An overfull graph is a graph whose number of its edges is greater than the product of its maximum degree and   2 n , where is the number of vertices. In this paper, some extremals of overfull graphs are presented. We also classify all plannar overfull graphs.


Introduction
All graphs in this paper are simple and denoted by .The k -edge coloring of a graph is an assignment of colors to the edges of the graph so that adjacent edges have different colors.The minimum required number of colors for the edges of a given graph is called the edge chromatic number of the graph and it is denoted by .In the next section, we compute some extremal overfull graphs and finally, in section three, we determinethe class of plannar overfull graph.Throughout this paper, our notation is standard and mainly taken from [1].

Results and Discussion
Let be the maximum degree of vertices of graph .Obviously, , and by Vizing's theorem .In other words, or .The graph is said to be of class 1 whenever, and otherwise, it is said to be of class 2.

 
. It is easy to see that the number of vertices of an overfull graph is an odd number and they are class 2. The following lemma, directly can be derived from the definition: Lemma 1.Every regular graph is overfull, where -r r is an even and is an odd integers.n The concept of overfull graph play a significant role in understanding of the edge chromatic properties of graphs.Chetwynd and Hilton [2] conjectured that a vaste category of graphs are class 2 if they contain an overfull subgraph with the same maximum degree: We know that this conjecture is solved under special conditions (see e.g.[3,4]).
The aim of this section is to compute the maximal and minimal overfull graphs.We show that trees and unicycle graphs are not overfull.In continuing, we compute the second, the third and the fourth extremal overfull graphs.
 and the proof is completed.
n An overfull graph is minimal if it has the minimum number of edges among all vertices overfull graphs and it is maximal if it has the maximum number of edges.
In the following theorem we find the minimal and maximal overfull graphs: , then among all vertices overfull graphs, the complete graph Let , the first claim is clear.For the second, since G is overfull then and so, .This implies that has a cycle.Clearly, is minimal overfull graph if and only if . By using Lemma 3, and so G is a cycle.
In Lemma 3, we classified the unicycle graphs on edges.In continuing, let be a graph with edges, since is overfull, thus edges, a contradiction.
Proof.By using Theorem 1, the proof of the first claim is clear.For the second part, note that 1 2 has a vertex of degree 2 and the others have degree 3. So, by Euiler Theorem, we have: has the minimum possible edges by this properties and this completes the proof.inim of degree 3, so by adding a new edge to it we have Proof.Since is plannar overfull graph, then G To find the the third m al overfull graph, note that the second minimal has  [3] T. Niessen, "How to Find Overfull Subgraphs in Graphs with Large Maximum Degree," Discrete Applied Mathematics, Vol. 51, No. 1-2, 1994, pp. 117-125.

Plannar Overfull Graphs
In this section, we classify all plan [4] M. Plantholt, "Overfull Conjecture for Graphs with High Minimum Degree," Journal of Graph Theory, Vol. 47, No. 2, 2004, pp. 73-80. doi:10.1002/jgt.20013this, we need followin lemma: The third minimal overfull graph with 7 n  vertices is a graph constructed by removing an edge from tisfies in the th dition, it is a 4-regular graph.Proof.The proofs of the first and second claims are  and so .Clearly, and we have the following cases: