Abstract

A graph is said to be even if all vertices of have even degree. Given a -edge-coloring of a graph , for each color let denote the spanning subgraph of in which the edge-set contains precisely the edges colored . A -edge-coloring of is said to be an -edge-coloring if for each color , is an even graph. A -edge-coloring of is said to be evenly-equitable if for each color , is an even graph, and for each vertex and for any pair of colors , . For any pair of vertices let be the number of edges between and in (we allow , where denotes a loop incident with ). A -edge-coloring of is said to be balanced if for all pairs of colors and and all pairs of vertices and (possibly ), . Hilton proved that each even graph has an evenly-equitable -edge-coloring for each . In this paper we extend this result by finding a characterization for graphs that have an evenly-equitable, balanced -edge-coloring for each . Correspondingly we find a characterization for even graphs to have an evenly-equitable, balanced 2-edge-coloring. Then we give an instance of how evenly-equitable, balanced edge-colorings can be used to determine if a certain fairness property of factorizations of some regular graphs is satisfied. Finally we indicate how different fairness notions on edge-colorings interact with each other.

1. Introduction

When considering edge-colorings of graphs it is usually desired to have some fairness properties imposed on the number of edges colored by each color. Below we define some important such notions and then explore the existence of edge-colorings satisfying combinations of these conditions.

In what follows, a graph is called even if all vertices of have even degree. Given a -edge-coloring of a graph , for each color let denote the spanning subgraph of in which the edge-set contains precisely the edges colored . Then a -edge-coloring of is called an even -edge-coloring if for each color , is an even graph. A -edge-coloring of is said to be equitable if for each vertex and for each pair of colors , . Moreover, a -edge-coloring of is said to be evenly-equitable if(i)for each color , is an even graph,(ii)for each vertex and for any pair of colors , .

For any pair of vertices , let be the number of edges between and in (we allow , where denotes a loop incident with ). A -edge-coloring of is said to be balanced if for all pairs of colors and and all pairs of vertices and (possibly ), . A -edge-coloring of is said to be equalized if for each pair of colors .

Due to de Werra’s work in [1–4] it has been known since the 1970s that for each every bipartite graph has a -edge-coloring that is balanced, equitable, and equalized at the same time. One important result for more general graphs is by Hilton, who proved in [5] that each even graph has an evenly-equitable -edge-coloring for each , thereby completely settling this problem (see Theorem 9). The existence of equitable -edge-colorings is much more problematic and very unlikely to be completely solved, since, for example, settling the existence of equitable -edge-colorings is equivalent to classifying Class 1 graphs (see [6, 7] for some results on this topic). One general result on equitable -edge-colorings was found by Hilton and de Werra [8] who proved that if and is a simple graph such that no vertex in has degree equal to a multiple of , then has an equitable -edge-coloring. More recently, Zhang and Liu [9] extended this result by proving that for each , if the subgraph of induced by the vertices which have degree equal to a multiple of is a forest, then has an equitable -edge-coloring, thereby verifying a conjecture made by Hilton in [10].

In Section 2 we extend Hilton’s result [5] by finding a characterization for graphs that have an evenly-equitable, balanced -edge-coloring for each (see Theorem 1). We then use this result to find a different kind of characterization for even graphs to have an evenly-equitable, balanced -edge-coloring (see Theorem 2).

In Section 3 we prove Theorems 5 and 6, the latter of which uses results from the previous section. The proof of Theorem 6 provides an instance of how evenly-equitable, balanced edge-colorings can be used to ensure that a certain fairness property of factorizations of some regular graphs is satisfied. This particular notion of fairness is defined as follows. A -factorization of a graph in which the vertices have been partitioned into parts is said to be fair if for each two parts (possibly they are the same) the number of edges between these two parts in each factor differs from the number in each other factor by at most one.

For completeness, in Section 4 we address the existence of all other combinations of the three edge-coloring properties (namely, evenly-equitable, balanced, and equalized), finding weakest subsets of conditions that will guarantee (if possible) that a graph has a -edge-coloring which has the following properties in turn: () evenly-equitable, balanced, and equalized, () evenly-equitable and equalized, () balanced and equalized, () evenly-equitable, () balanced, and () equalized.

For each proper subset of the vertex set of a graph , define the edge-cut . Let be such that (modulo ). Let be the spanning subgraph of in which for each pair of vertices and the number of edges between and is . Then clearly (modulo ) for all . For the purposes of this paper, a vertex is said to be odd (even) if is an odd (even) integer.

2. Coloring Results

The following characterization can be used to find evenly-equitable, balanced -edge-colorings. The proof has the flavor of Hilton’s proof in [5] of the case where the additional property of being balanced was not required but is modified to deal with extra complications that arise in this new setting.

Theorem 1. For each positive integer , a graph (possibly with loops) has an evenly-equitable, balanced -edge-coloring if and only if it has an even, balanced -edge-coloring.

Proof. Proving the “only if” result is trivial. To show the “if” result, we first prove the assertion for the case when is connected and loopless. Let be an even, balanced -edge-coloring of . Among all pairs of colors and all vertices suppose that is as large as possible (where ). If , then this edge-coloring is evenly-equitable, so assume . Let be the spanning subgraph of induced by the edges colored and . From form a new graph by adding an uncolored loop at each vertex satisfying (mod 4). Then For each pair of vertices with and for any color , let , and let be a set of size containing precisely edges of each of the colors and joining vertices and . So is even. Let and . Define . Since is even for each , and since the original edge-coloring is even, each component of is an eulerian graph and has no multiple edges since is balanced (possibly it has an uncolored loop at some of the vertices). The following argument establishes property (4); namely, each component of has an even number of edges. First note that by the assumption of this theorem for all each component of is eulerian, so Let be any component of and let . Let ; so is even (since there are an even number of edges in between each pair of vertices). Let be the edge-cut , which by the definition of satisfies . So, . Furthermore, since for each color and are edge-cuts in and , respectively, by (2)   and are even. Hence (mod 4). Then,So, Let be a new -edge-coloring of formed as follows. For each component of , alternately color the edges of an eulerian circuit of with and . This yields a balanced -edge-coloring of ( is simple) where by (4), for each vertex , Now add the edges in with their original colors to and remove the uncolored loops that were added when forming . Then clearly the resulting graph is and this new -edge-coloring satisfies for each . To show that is also even, consider the following cases (in which refers to edge-coloring with ).
Case 1. One has (mod 4). Note that in this case we are not adding a loop at when forming . Now look at the following subcases.
Subcase 1.1. is odd. So, an odd number of edges incident with of each of the colors and were removed when forming from . So, (mod 4) and hence by (5)   (mod 2). Putting back the removed edges shows that is incident with an even number of edges of each color in the edge-coloring of .
Subcase 1.2. is even. So, an even number of edges incident with of each of the colors and were removed when forming . So, (mod 4) and hence (mod 2). Putting back the removed edges shows that is incident with an even number of edges of each color in the edge-coloring of .
Case 2. One has (mod 4). Note that in this case an uncolored loop is added to when forming . Now look at the following subcases.
Subcase 2.1. is odd. So, after adding an uncolored loop at , an odd number of edges incident with of each of the colors and were removed when forming . Then (mod 4), so by (5) in the new edge-coloring (mod 2). So, for each and each (mod 2).
Subcase 2.2. is even. So, after adding an uncolored loop at , an even number of edges incident with of each of the colors and were removed when forming . Then (mod 4), so by (5) in the new edge-coloring (mod 2). So, for each and each , (mod 2).
Repetition of this procedure yields an evenly-equitable, balanced -edge-coloring of .
For the case when has loops and is possibly disconnected, simply remove all the loops from and apply this procedure to each component of the resulting loopless graph to get an evenly-equitable, balanced -edge-coloring of each component. Then put back the loops; it is easy to color them in a balanced way without destroying the evenly-equitable property at each vertex.

Note that in the statement of Theorem 1 we cannot replace the condition on the existence of an even, balanced -edge-coloring by a weaker set of conditions, as is illustrated by the next two examples. A cycle of length with a cycle of length intersecting in one of its vertices is an even graph and clearly has a balanced (and equalized) -edge-coloring, but no -edge-coloring that is evenly-equitable and balanced. The graph (the graph with two vertices and two edges joining these two vertices) has an even (actually evenly-equitable) -edge-coloring, but no -edge-coloring that is evenly-equitable and balanced. While these two graphs are trivial, they can be generalized to more complicated examples.

Theorem 1 leads to the problem of finding conditions guaranteeing that a graph has an even, balanced -edge-coloring. The following result addresses that problem. Recall that our unusual definitions of even and odd vertices and of are given at the end of Section 1.

Theorem 2. has an even, balanced -edge-coloring if and only if is even and has no components with an odd number of odd vertices.

Proof. To prove the necessity, suppose that an even, balanced -edge-coloring of is given. Since the given -edge-coloring is balanced, for each pair of vertices and , the edges between and that are to be deleted when forming from can be chosen so that they are shared evenly among the two color classes. Let be a component in . Now since the given -edge-coloring of is even, for each color , an odd vertex in contributes an odd number to the degree sum of the graph , and an even vertex in contributes an even number to the degree sum of the graph . Hence the number of odd vertices in must be even.
To show the sufficiency, color the edges in as follows. To satisfy the balanced property, for each pair of vertices color (note that by definition of this is an integer) of the edges between and with each color . Let be the graph induced by the edges that have been colored so far, and note that the graph induced by the uncolored edges is . Also note that by the definition of odd and even vertices, for each , Since is an even graph and since is even for each , is also an even graph. For each component in color the edges of an eulerian tour of as follows. Start by coloring the first edge in the eulerian tour with and then switch to (modulo 2) whenever the eulerian tour reaches an odd vertex for the first time. Note that if the first vertex in the eulerian tour is even, then the first and last edges in the eulerian tour will have the same color because an even number of color switches will occur (by assumption there are an even number of odd vertices). Similarly, if the first vertex, say , is odd, then the first and the last edges will have different colors if (since no color switch is made at ) and they will have the same color if (since then the eulerian tour will pass through , so a color switch will occur at ). This coloring of the edges in has the property that for each and for each (i)if is odd, then is odd,(ii)if is even, then is even. So, for each and each , since , by , (i), and (ii) each vertex in has even degree and hence the given -edge-coloring has the desired properties.

It appears to us that a generalization of Theorem 2 for three or more colors may be difficult to obtain.

The following result characterizes graphs which have an evenly-equitable, balanced -edge-coloring.

Corollary 3. Suppose that is an even graph. Then has an evenly-equitable, balanced -edge-coloring if and only if has no components with an odd number of odd vertices.

Proof. This follows immediately by Theorems 1 and 2.

3. An Application Using Amalgamations

In this section edge-colorings that satisfy another notion of equally distributing edges across color classes are considered, namely, that of fairness. Not only are the edge-colorings equitable, but also for any given partition of the vertices, for each two parts in (possibly they are the same) the edges between vertices in the two parts are equally divided among the color classes. While the results here (Theorems 5 and 6) address general partitions, these types of questions naturally arise when edge-coloring the complete multipartite graph , in which the partition is chosen to be the parts of the graph. For example, it has been shown when there exist fair equitable edge-colorings of in which each color class induces a hamilton cycle [11] or a -factor [12].

To prove Theorem 5, the method of amalgamations is used. A graph is said to be the -amalgamation of a graph if is a function from onto such that if and only if . The function is called an amalgamation function. We say that is a detachment of , where each vertex of splits into the vertices of . An -detachment of is a detachment in which each vertex of splits into vertices. Amalgamating a finite graph to form the corresponding amalgamated graph can be thought of as grouping the vertices of and forming one supervertex for each such group by squashing together the original vertices in the same group. An edge incident with a vertex in is then incident with the corresponding new vertex in ; in particular an edge joining two vertices from the same group becomes a loop on the corresponding new vertex in .

In what follows, denotes the subgraph of induced by the edges colored (so unlike , is not necessarily a spanning subgraph), and denotes the number of loops at in . The following theorem was proved in much more generality by Bahmanian and Rodger in [13], but this is sufficient for our purposes.

Theorem 4. Let be a -edge-colored graph and let be a function from into such that for each , implies . Then there exists a loopless -detachment of with amalgamation function , being the number function associated with , such that satisfies the following property: (i) for each and each and each ,(ii) for each with and every pair of distinct vertices ,(iii), for every pair of distinct vertices , each , and each .

The following theorem provides a necessary condition for the existence of fair 2-factorizations of -regular graphs (). For any graph and any partition of , let be the -amalgamation of , where maps two vertices in to the same vertex in if and only if they are in the same element of .

Theorem 5. Let be a -regular graph . Let be any partition of . Let . Suppose that has a fair -factorization. Then (1) has no components with an odd number of odd vertices.

Proof. Suppose that has a fair -factorization. Let and be the subgraphs of induced by the edges corresponding to the -factors of . Since at each vertex in the number of edge-ends incident with a vertex is a multiple of and since these edge-ends are shared evenly among and , the number of edge-ends incident with each vertex in in each of and is even. So, by the definition of odd and even vertices, in an odd vertex is incident with an odd number of edge-ends in each of and , and an even vertex is incident with an even number of edge-ends in each of and . Let be a component of . Clearly is an even number and where is an even number and each term in the summation is an odd number by the above observation. Hence the number of odd vertices in must be even.

To investigate whether the necessary condition given in Theorem 5 is also sufficient for a graph to have a fair -factorization, we introduce the notion of -equivalence.

Let and be two graphs with , and let be a partition of . Then is said to be -equivalent to if for all (possibly ) , where denotes the number of edges in (for ) between the parts and . So if and are -equivalent, then . If either or has a fair -factorization, then Theorem 5 shows that must be satisfied. To investigate the strength of , Theorem 6 shows that if is a -regular graph for which satisfies , then is -equivalent to some graph (which is simple if a certain necessary condition is met) with a fair -factorization. Conjecture 7 goes on to make a much stronger claim that if is -equivalent to , then has a fair -factorization if and only if does.

Theorem 6. Let be a -regular graph. Let be any partition of . Let . Suppose has no components with an odd number of odd vertices. Then there exists a graph such that(i),(ii) is -equivalent to ,(iii) has a fair -factorization (with respect to the given partition ),(iv) can be chosen to be simple if and only if for all , if , and if .

Note that it is long known by Petersen’s -factor theorem (see, e.g., [14]) that every -regular graph has a -factorization. The importance of Theorem 6 is that if the condition of the theorem is satisfied, then regardless of the partition that is chosen, the resulting factorization of (formed with in mind) is fair.

Proof. By the supposition has no components with an odd number of odd vertices. Clearly is even since is even. So satisfies the conditions of Corollary 3 and hence it has an evenly-equitable, balanced -edge-coloring. By the evenly-equitable property of this -edge-coloring, each color appears on exactly half of the edge-ends incident with each vertex of (a loop contributes two edge-ends to the incident vertex). Notice that is the -amalgamation of where if and only if and are in the same element of . For each define . By (i) of Theorem 4, there exists an -detachment of such that (1) is -equivalent to ,(2)for each vertex of the edges of each color incident with are shared as evenly as possible among the vertices in (i.e., the vertices in the corresponding part of ). Note that, by (ii) and (iii) of Theorem 4, will be simple if for all , if , and if . Clearly these are necessary conditions if the -detachment of is to be simple.
By (2), in each color is on two edges incident with each vertex. So, in the subgraph induced by the edges of each color is a -factor, and hence this -edge-coloring is a -factorization of . The fairness of this -factorization follows from the following observation: There is a one-to-one correspondence between the edges colored joining any pair of vertices and in and the edges colored between the two corresponding parts and of . So, the balanced property of this -edge-coloring implies the required fairness property of the -factorization.

In the light of Theorems 5 and 6 we make the following conjecture.

Conjecture 7. Let be a -regular graph . Let be any partition of . Let . Suppose has no components with an odd number of odd vertices. Then has a fair -factorization.

4. Other Combinations of Requirements

As described in the introduction we now consider other combinations of edge-coloring properties in turn. The results in this section are straight forward to obtain but are reported here for completeness.

() Evenly-equitable, balanced, and equalized: as is discussed below, the examples in Figure 1 show that there are graphs which have an even, balanced, equalized -edge-coloring, but no -edge-coloring that is evenly-equitable and equalized. So, for each positive integer , no matter which combination of the conditions on the existence of an even -edge-coloring, balanced -edge-coloring and equalized -edge-coloring of a graph is used, it is not possible to guarantee that has a -edge-coloring which is evenly-equitable, balanced, and equalized.

(a) : A vertex-minimum example

(b) : An edge-minimum example

(a) : A vertex-minimum example
(b) : An edge-minimum example

Figure 1 

Examples of graphs that are not of color-type 1.

A graph is said to be of color-type 1 if it is connected and simple and has an even, equalized -edge-coloring but has no evenly-equitable, equalized -edge-coloring. Note that any edge-coloring of a color-type 1 graph is balanced because it is simple. In there are two -cycles that intersect in just the top vertex; color the six edges in these -cycles with color and color the remaining edges with color to produce an even, balanced, equalized -edge-coloring. does not have an evenly-equitable, equalized -edge-coloring, since in every evenly-equitable -edge-coloring one color class must be -regular and spanning and so has edges. So, is of color-type . In fact, a basic search shows that there is no color-type graph with fewer vertices nor one on 7 vertices with less than edges.

In the six edges of the two -cycles can be colored with color and the edges of the -cycle with color , thereby producing an even, balanced, equalized -edge-coloring. does not have an evenly-equitable, equalized -edge-coloring, since the only evenly-equitable -edge-coloring has one color class consisting of the three edges in the middle -cycle. So, is of color-type . In fact, another basic search shows that there is no color-type graph with fewer edges nor one with 11 edges on less than 9 vertices.

Note that suggests a way to construct infinitely many color-type 1 graphs: Take any cycle of length as the middle cycle, attach to it a cycle of length on the left and a cycle of length on the right where , and .

Since we cannot guarantee the existence of an evenly-equitable, balanced, and equalized -edge-coloring of a graph , even with the strong assumption that has a -edge-coloring which is even, balanced, and equalized, we focus our attention on conditions implying the existence of -edge-colorings that are () evenly-equitable and equalized, () balanced and equalized, () evenly-equitable, () balanced, and () equalized; evenly-equitable, balanced edge-colorings are the focus of Section 2.

() Evenly-equitable and equalized: the examples in Figure 1 show that even with the strong assumption that a graph has an even, balanced, equalized -edge-coloring, does not necessarily have an evenly-equitable, equalized -edge-coloring; characterizations of graphs with such edge-colorings would seem to be difficult to find.

() Balanced and equalized: such edge-colorings are always easy to find as is stated in the following theorem.

Theorem 8. For each positive integer , each graph has a balanced, equalized -edge-coloring.

Proof. Let be a graph with edges (loops, being special types of edges, are also included in this count). Form an ordering of the edges of where loops incident with the same vertex appear consecutively in the list, as do the edges joining the same pair of vertices. For color with (modulo ). This -edge-coloring is clearly balanced and equalized.