This paper provides structural characterization of simple graphs whose edge set can be partitioned into maximum matchings. We use Vizing's classification of simple graphs based on edge chromatic index.

By a simple graph, we shall mean a graph with no loop and no multiple edges. We will only consider simple graphs with

We now consider simple graphs whose edge set can be partitioned into maximum matchings. Complete graphs and even cycles are some of the examples but there are numerous other examples too. For instance, consider the graph in Figure

A graph whose edge set can be partitioned into maximum matchings.

Vizing’s celebrated theorem states that

Our main aim in this paper is to prove the following results.

Let

If

If

We first establish some basic results that will be extremely useful in the next section. We will be borrowing some ideas and results discussed in [

Let

For a simple graph

It is obvious that for each graph

Note that the above bound can also be inferred from [

A simple, connected graph

Let

We will consider graphs with

For

We next consider all cases involving

Let

We next prove that

Let

Let

If the statement (a) is false then there exists a vertex

On the contrary assume that

The above inequality implies that

Since the statements

If

If

Therefore there is a unique vertex

We emphasize that graphs considered in this discussion have no isolated vertex. Note that a method is provided in [

Let

Let

If the statement (a) is false then there exists a vertex

We again note that the edge bound, that is, (

We recall that the graph

Let

Note that in the following inequality we again used the fact that the edge bound, that is, (

Let

Now since

From (b) and (c), every component

This follows from (d).

Now we explore the inverse of Proposition

Let

We use the method given in [

We can combine the above two propositions in the following theorem.

Let

The above conclusion follows by Propositions

A simple graph

Examples:

We observe that a proper edge coloring of

We consider a minimal proper edge coloring of

Suppose

If

Any proper edge coloring of

The name “friendly-edge-colorable” is due to Proposition

The following theorem characterizes friendly-edge-colorable graphs in Class II.

Let

every component of

Let

Now we will consider friendly-edge-colorable graphs that are in Class I, that is, graphs with edge chromatic index

Let

As

Lemma

Let

As

By a nontrivial component of a simple graph, we shall mean a component that has at least an edge, that is,

We first show the if part. Consider a proper edge coloring for each component in

Next we show the only if part. Since

If for some component

In the previous lemma the condition that

If

factor critical, friendly-edge-colorable graphs with edge chromatic index

Let

Reader can review Figure

The author would like to thank Dr. Nishali Mehta, Dr. Naushad Puliyambalath, and Professor Ákos Seress for their valuable comments and help to improve the paper.