Abstract
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.
1. Introduction
By a simple graph, we shall mean a graph with no loop and no multiple edges. We will only consider simple graphs with no isolated vertex. We first fix some notations. For a graph , , and would denote the edge set and the vertex set of , respectively. , , and would denote the maximum degree of any vertex in , the size of a maximum matching in and the edge chromatic index of , respectively. For , would denote the degree of the vertex and would denote the induced subgraph on .
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 1.
Vizing’s celebrated theorem states that for a simple graph . The definition of the edge chromatic index implies that . Therefore Vizing classified simple graphs as follows: a simple graph is in class I if and only if and a simple graph is in class II if and only if . There is no structural characterization yet known for the graphs in class I or in class II. It is NP-complete to determine whether a simple graph is in Class I or Class II (see [1]). But under certain restrictions structural characterization of class I and class II graphs has been achieved. It is also known that all planar graphs with maximum degree at least seven are in Class I (see [2, 3]). Another interesting result concerns itself with relative cardinality of Class I and Class II (see [4]). We will characterize Class I and class II graphs whose edge set can be partitioned into maximum matchings.
2. Results
Our main aim in this paper is to prove the following results.
Theorem 1. Let be a simple graph such that , , and . is a unique graph up to isomorphism if and only if divides .
Theorem 2. If is a Class II graph and has a partition into maximum matchings, then is even and is the graph with exactly components each isomorphic to , the complete graph of order .
Theorem 3. If is a Class I graph and can be partitioned into maximum matchings, then has a partition into subgraphs that are either or a factor critical graph such that can also be partitioned into maximum matchings and .
3. Preliminaries
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 [5]. The following definition is in [5].
Definition 4. Let , be positive integers and be a simple graph with and . if and only if for any simple graph such that and is a subgraph of , either or .
Theorem 5. For a simple graph , if and , then
Proof. It is obvious that for each graph with and there exists a maximal graph such that , , and . Note that a maximal graph such as can be constructed by adding edges or new vertices and edges to . The upper bound for obtained in [5] implies that for any , . Therefore,
Note that the above bound can also be inferred from [6]. However, the method used in [6] is more involved and it does not help in ascertaining uniqueness of the graphs achieving the edge bound (1) for given and values, that is, Theorem 1. Our first goal is to prove Theorem 1. We next define a factor-critical graph and state the Gallai’s lemma that is crucial to the following discussion. An elegant proof of the Gallai’s lemma can be found in [7].
Definition 6. A simple, connected graph is called factor-critical if and only if has a perfect matching for all .
Lemma 7 7 (Gallai). Let be a simple, connected graph. If for all , then is a factor-critical graph.
We will consider graphs with no isolated vertex only. Now we consider those graphs that attain the edge bound and analyze under what conditions these graphs are unique. We first consider some trivial cases.
Remark 8. For , the graph that attains the edge bound (1) consists of components where each component is . For and , the unique graph that attains the edge bound (1) is . For and the unique graph that attains the edge bound is . But there are two graphs that satisfy , and attain the edge bound . These graphs are and .
We next consider all cases involving and .
3.1. Unique Graph
Let be a positive integer. We consider simple graphs with no isolated vertex such that Note that the last equation ensures that attains the maximum edge bound given by (1). We will now construct a graph satisfying (4) as follows.
Case (I). Let be an even integer. In this case, let .
Case (II). Let for some . In this case, let be the graph obtained from by removing a maximum matching. To obtain , we connect any of the vertices of to a new vertex, .
We next prove that is the unique graph satisfying (4).
Proposition 9. Let and be the simple graph described above. If is a simple graph satisfying (4) then(a) for all ,(b) is connected,(c).
Proof. Let be a graph satisfying the conditions of the proposition.
Proof of (a). If the statement (a) is false then there exists a vertex such that . As at most one edge can cover in any maximum matching, we have . Therefore, The above expression is a nondecreasing function of for a fixed . Since , we have Therefore by assumption and the above equation, we have . But , since . Hence the statement (a) holds.
Proof of (b). On the contrary assume that is not connected. Let be a component of . Then as has no isolated vertex and is not connected by assumption. By statement (a) and Gallai’s Lemma (Lemma 7), is a factor-critical component. Therefore, . So,
The above inequality implies that
So there is a component of such that as . But the (2) demands that . The contradiction implies that the statement (b) holds.
Proof of (c). Since the statements and hold for , is factor-critical by Gallai’s Lemma (Lemma 7). As , we have . We consider following two cases.
If is even then is a connected graph with vertices and . Therefore, for all . Hence is , the complete graph on vertices. So .
If is odd. Let for some . Then , and . So
Therefore there is a unique vertex of degree . Hence there is a vertex in which is not a neighbor of . Consequently is a regular graph of degree on vertices and hence its complement is a regular graph of degree one, namely, a matching of a complete graph on vertices. This establishes , where is the graph described earlier for the case .
3.2. Unique Graphs with Maximum Number of Edges for a Given Maximum Degree and Matching Size
We emphasize that graphs considered in this discussion have no isolated vertex. Note that a method is provided in [5] to construct a graph such that attains the edge bound given by (1). We find the condition when the graphs that attain the maximum edge bound (1) are unique up to isomorphism.
Proposition 10. Let be the graph constructed in the Section 3.1 and be a simple graph with and such that . If divides and , then (a) for all ,(b)if is a component of with , then ,(c)if is a component of with , then , (d)every component of is isomorphic to ,(e) is unique up to isomorphism.
Proof. Let be a graph satisfying the conditions of the proposition.
Proof of (a). If the statement (a) is false then there exists a vertex such that . This implies
We again note that the edge bound, that is, (1) is a nondecreasing function of for a fixed . Also . Therefore,
This contradiction proves (a).
We recall that the graph is a factor critical graph with edges and maximum matching size .
Proof of (b). Let be a component of . Gallai’s lemma and (a) imply that is factor-critical, hence and
Note that in the following inequality we again used the fact that the edge bound, that is, (1) is a nondecreasing function of for a fixed . If , then
This proves (b).
Proof of (c). Let be a component of . By Gallai’s lemma and (a), is a factor-critical graph and hence . Thus
Now since , we have
which proves (c).
Proof of (d). From (b) and (c), every component of has and also . Since is the only graph with maximum matching size and the number of edges , must be isomorphic to . This proves (d).
Proof of (e). This follows from (d).
Now we explore the inverse of Proposition 10. Recall the connected graph described in Section 3.1 is the unique graph satisfying (4).
Proposition 11. Let be a simple graph such that , that is, attains the maximum edge bound given by the inequality (1). If , and does not divide , then there exists a simple graph such that , , and is not isomorphic to .
Proof. We use the method given in [5] to construct a simple graph such that , , and . Let have components isomorphic to and components isomorphic to (described in Section 3.1). Let if is not isomorphic to . So assume that is isomorphic to . Note that as does not divide . If then let and be two components of isomorphic to . We remove an edge of and then connect the vertex of degree of to any vertex of degree one in by a new edge. Thus, a new graph is obtained. It is obvious by construction that satisfies hypothesis of Proposition 11. So, we need to consider only the case to complete the proof. As and , has at least two components. Hence has a component isomorphic to and at least one component isomorphic to . As , we can coalesce the two components to form a factor-critical component with vertices and maximum degree at most which has number of edges equal to and maximum matching size equal to . Thus, again a new graph is obtained. It is obvious by construction that satisfies hypothesis of Proposition 11.
We can combine the above two propositions in the following theorem.
Theorem 12. Let be a simple graph such that , and . is a unique graph up to isomorphism if and only if divides .
Proof. The above conclusion follows by Propositions 10 and 11.
4. Graphs Whose Edge Set Can Be Partitioned into Maximum Matchings
Definition 13. A simple graph is called friendly-edge-colorable if and only if has a partition into maximum matchings.
Examples: , for , for , for and for .
We observe that a proper edge coloring of is equivalent to partitioning into matchings (may not be maximum).
Proposition 14. is a friendly-edge-colorable graph if and only if .
Proof. We consider a minimal proper edge coloring of . Since every color class of is a matching and a matching in can be of size at most , we have . So if is a friendly-edge-colorable graph, then there is a partition of into maximum matchings. Hence there is a positive integer such that . This partition corresponds to a proper edge coloring with colors. Thus, . Hence, . Thus for a friendly-edge-colorable graph , we have .
Suppose is not a friendly-edge-colorable graph. We consider an edge coloring of in colors. Note that each color class is a matching and at least one of the color classes is of size strictly less than as is not a friendly-edge-colorable graph. Thus, we get .
Proposition 15. If is a friendly-edge-colorable graph, then any proper edge coloring of in colors results in a partition of into maximum matchings.
Proof. Any proper edge coloring of is a partition of into matchings (may not be maximum). Consider a proper edge coloring of in colors. If there is a color class of size strictly less than , then which contradicts Proposition 14.
Remark 16. The name “friendly-edge-colorable” is due to Proposition 15 which states that when the least number of colors are used to properly color the edges, the colors are equally distributed so that each color class gets the maximum!
The following theorem characterizes friendly-edge-colorable graphs in Class II.
Theorem 17. Let be a friendly-edge-colorable graph of Class II such that and . Then(a) is even,(b) divides ,(c)every component of is isomorphic to , the complete graph of order .
Proof. Let and . By Proposition 15, . Also by inequality (1), Thus, , which proves . Also, that implies (). By Proposition 10, we know that every component of is isomorphic to , the complete graph of order .
Now we will consider friendly-edge-colorable graphs that are in Class I, that is, graphs with edge chromatic index . Next we prove three lemmas that help us characterize friendly-edge-colorable graphs in Class I.
Lemma 18. Let be a friendly-edge-colorable graph. If there exists a vertex such that then .
Proof. As is friendly-edge-colorable, there exists a partition of into maximum matchings and, by Proposition 14, . Hence each class, that is, each part of this partition, has size and there are parts. We denote each color class (i.e., a part in the partion) by for all . If then there is a color missing at . Without loss of generality, let the missing color be . Hence the whole color class belongs to , that is, . Therefore, . But is a maximum matching of hence . This contradicts that . Hence . Also by the definition of .
Remark 19. Lemma 18 implies that if is friendly-edge-colorable and there exists such that , then as . Hence is in Class I. So the contrapositive implies that if is friendly-edge-colorable and in Class II, that is, not in Class I, then for all . By Gallai’s lemma, each component of is a factor-critical component as we noticed in Theorem 17.
Lemma 20. Let be a friendly-edge-colorable graph. If there exists such that and , then is a friendly-edge-colorable graph and .
Proof. As is friendly-edge-colorable, has a partition into maximum matchings. Define a coloring, , corresponding to such a partition. Then each color class has size and there are color classes by Proposition 14. Now consider the restriction of on . By Lemma 18, hence has color classes each of size . As by assumption , has a partition into maximum matchings, implying that is friendly-edge-colorable. Since is friendly-edge-colorable, by Proposition 14, we get
By a nontrivial component of a simple graph, we shall mean a component that has at least an edge, that is, is a subgraph of the component.
Lemma 21. is a friendly-edge-colorable graph if and only if each nontrivial component of is friendly-edge-colorable and .
Proof. We first show the if part. Consider a proper edge coloring for each component in colors. This gives the desired partition of the edge set of .
Next we show the only if part. Since is friendly-edge-colorable, there is a partition of into maximum matchings that has classes (by Proposition 14). We claim that restricting this partition to any component of gives a partition of the edge set of the component into maximum matchings of the component. Suppose on the contrary that some matching obtained this way (by restricting a maximum matching to ) for a component is not a maximum matching in . Then there is an augmenting path relative to this matching in the component. But then this is an augmenting path, relative to the corresponding matching of , in . Hence is not a maximum matching of . This contradicts the fact that is a maximum matching.
If for some component , then (as for is a component of ). Now consider any proper edge coloring of in colors and for each component order color classes greedily. For each , we denote the th color class by . Then the first color classes will have strictly larger size than the remaining color classes . This contradicts that each color class is of the same size as shown by Proposition 15.
Remark 22. In the previous lemma the condition that for all components is necessary otherwise counter examples exist. For instance, let be the graph consisting of two components one of which is and the other one is .
Theorem 23. If is a nontrivial, Class I, friendly-edge-colorable graph then has a partition into the following two kinds of subgraphs:(i)(ii)factor critical, friendly-edge-colorable graphs with edge chromatic index .
Proof. Let . If there exists such that then remove all edges incident to and define . Since by Lemma 18 and Remark 19 , we removed a from . Note that is a friendly-edge -colorable graph and by Lemma 20. By Lemma 21, each component of is also a friendly-edge-colorable graph and has edge chromatic index . Similarly, for define if there exists such that . We remove the vertices and all edges adjacent to each of these vertices. Corresponding to each of the ’s, we get a subgraph isomorphic to . For some large enough , we have only those nontrivial components in such that for all . By Gallai’s lemma, these are the factor critical components. Also by Lemma 21, .
Remark 24. Reader can review Figure 1 and notice that the degree of the vertex in Figure 1 is and must be covered by every maximum matching. Also, removal of the vertex from the graph yields a graph whose only nontrivial component is factor-critical.
Acknowledgments
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.