# On maximum k-edge-colorable subgraphs of bipartite graphs

If k≥ 0, then a k-edge-coloring of a graph G is an assignment of colors to edges of G from the set of k colors, so that adjacent edges receive different colors. A k-edge-colorable subgraph of G is maximum if it is the largest among all k-edge-colorable subgraphs of G. For a graph G and k≥ 0, let ν_k(G) be the number of edges of a maximum k-edge-colorable subgraph of G. In 2010 Mkrtchyan et al. proved that if G is a cubic graph, then ν_2(G)≤|V|+2ν_3(G)/4. This result implies that if the cubic graph G contains a perfect matching, in particular when it is bridgeless, then ν_2(G)≤ν_1(G)+ν_3(G)/2. One may wonder whether there are other interesting graph-classes, where a relation between ν_2(G) and ν_1(G)+ν_3(G)/2 can be proved. Related with this question, in this paper we show that ν_k(G) ≥ν_k-i(G) + ν_k+i(G)/2 for any bipartite graph G, k≥ 0 and i=0,1,...,k.

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## 1 Introduction

In this paper graphs are assumed to be finite, undirected and without loops, though they may contain multiple edges. The set of vertices and edges of a graph is denoted by and , respectively. The degree of a vertex of is denoted by . Let and be the maximum and minimum degree of a vertex of . A graph is regular, if . The girth of the graph is the length of the shortest cycle in its underlying simple graph.

A bipartite graph is a graph whose vertices can be divided into two disjoint sets and , such that every edge connects a vertex in to one in . A graph is nearly bipartite, if it contains a vertex, whose removal results into a bipartite graph.

A matching in a graph is a subset of edges such that no vertex of is incident to two edges from the subset. A maximum matching is a matching that contains the largest possible number of edges.

If , then a graph is called -edge colorable, if its edges can be assigned colors from a set of colors so that adjacent edges receive different colors. The smallest integer , such that is -edge-colorable is called chromatic index of and is denoted by . The classical theorem of Shannon states that for any graph Shannon:1949 ; stiebitz:2012 . On the other hand, the classical theorem of Vizing states that for any graph stiebitz:2012 ; vizing:1964 . Here is the maximum multiplicity of an edge of . A graph is class I if , otherwise it is class II.

If the edges of are colored, then for a color let be the set of edges of that are colored with . Observe that is a matching. We say that a vertex is incident to the color , if is incident to an edge from . If is not incident to the color , then we say that misses the color . Now, if we have two different colors and , then consider the subgraph of induced by . Observe that the components of this subgraph are paths or even cycles. The components which are paths are usually called -alternating paths or Kempe chains stiebitz:2012 . If is an -alternating path connecting the vertices and , then we can exchange the colors on and obtain a new edge-coloring of . Observe that if is incident to the color in the former edge-coloring, then in the new one it will miss the color .

If , we cannot color all edges of with colors. Thus it is reasonable to investigate the maximum number of edges that one can color with colors. A subgraph of a graph is called maximum -edge-colorable, if is -edge-colorable and contains maximum number of edges among all -edge-colorable subgraphs. For and a graph let

 νk(G)=max{|E(H)|:H is a k-edge-colorable subgraph of G}.

Clearly, a -edge-colorable subgraph is maximum if it contains exactly edges.

There are several papers where the ratio has been investigated. Here is a maximum -edge-colorable subgraph of . bollobas:1978 ; henning:2007 ; nishizeki:1981 ; nishizeki:1979 ; weinstein:1974 prove lower bounds for the ratio when the graph is regular and . For regular graphs of high girth the bounds are improved in flaxman:2007 . Albertson and Haas have investigated the problem in haas:1996 ; haas:1997 when is a cubic graph. See also samvel:2010 , where the authors proved that for every cubic graph and . Moreover, samvel:2014 shows that for any cubic graph .

Bridgeless cubic graphs that are not -edge-colorable are usually called snarks cavi:1998 , and the problem for snarks is investigated by Steffen in steffen:1998 ; steffen:2004 . This lower bound has also been investigated in the case when the graphs need not be cubic in miXumbFranciaciq:2013 ; Kaminski:2014 ; Rizzi:2009 . Kosovski and Rizzi have investigated the problem from the algorithmic perspective Kosovski:2009 ; Rizzi:2009 . Since the problem of constructing a -edge-colorable graph in an input graph is NP-complete for each fixed , it is natural to investigate the (polynomial) approximability of the problem. In Kosovski:2009 , for each an algorithm for the problem is presented. There for each fixed value of , algorithms are proved to have certain approximation ratios and they are tending to as tends to infinity.

Some structural properties of maximum -edge-colorable subgraphs of graphs are proved in samvel:2014 ; MkSteffen:2012 . In particular, there it is shown that every set of disjoint cycles of a graph with can be extended to a maximum -edge colorable subgraph. Also there it is shown that a maximum -edge colorable subgraph of a simple graph is always class I. Finally, if is a graph with girth and is a maximum -edge colorable subgraph of , then and the bound is best possible is a sense that there is an example attaining it.

In samvel:2010 Mkrtchyan et al. proved that for any cubic graph . For bridgeless cubic graphs, which by Petersen theorem have a perfect matching, this inequality becomes, . One may wonder whether there are other interesting graph-classes, where a relation between and can be proved. In lianna:2017 , the following conjecture is stated:

###### Conjecture 1.

(lianna:2017 ) For each and a nearly bipartite graph

 νk(G)≥⌊νk−1(G)+νk+1(G)2⌋.

In the same paper the bipartite analogue of this conjecture is stated, which says that for bipartite graphs the statement of the Conjecture 1 holds without the sign of floor. Note that lianna:2017 verifies Conjecture 1 and its bipartite analogue when contains at most one cycle.

The present paper is organized as follows: In Section 2, some auxiliary results are stated. Section 3 proves the main result of the paper, which states that for any bipartite graph , , where . Section 4 discusses the future work.

Terms and concepts that we do not define, can be found in west:1996 .

## 2 Auxiliary results

In this section, we present some auxiliary results that will be useful later. The first two of them are simple consequences of a classical theorem due to König stiebitz:2012 ; west:1996 , which states for any bipartite graph , we have .

###### Proposition 1.

Let be a bipartite graph and let . Then a subgraph of is -edge-colorable, if and only if .

###### Proposition 2.

Let and let be a -regular bipartite graph. Then for we have .

Our next auxiliary result follows from an observation that a vertex can be incident to at most edges in a -edge-colorable graph.

###### Proposition 3.

If is a graph, is a vertex of and . Then

 νk(G)≤νk(G−v)+k.

The next result states that if one is removing an edge from a graph, then can decrease by at most one.

###### Proposition 4.

If is a graph, is an edge of and . Then

 νk(G−e)≤νk(G)≤νk(G−e)+1.

In order to prove our next auxiliary result, we will use alternating paths.

###### Lemma 1.

Let be a bipartite graph, be an edge of , and . Then for any maximum -edge-colorable subgraph with , we have or .

###### Proof.

Assume that there is a maximum -edge-colorable subgraph that does not contain and with and . Then there are colors and of such that misses at and misses at . Clearly, must be present at and must be present at , since is maximum -edge-colorable. Consider the alternating paths starting at and

. If they are the same, then we get an odd cycle contradicting the fact that

is bipartite. Hence they are different. Exchange the colors and on one of them and color . Observe that we have got a -edge-colorable subgraph of with edges contradicting the maximality of . Thus the statement of the lemma should be true. ∎

If is a matching in a graph , then a simple odd path is said to be -augmenting, if the odd edges of lie outside , the even edges of belong to , and the end-points of are not covered by . It is easy to see that if contains an -augmenting path, then is not a maximum matching in . The classical theorem of Berge berge:1973 , states that if is not a maximum matching in , then must contain an -augmenting path. In the end of this section, we prove the analogue of this result for -edge-colorable subgraphs of bipartite graphs. It is quite plausible that our result can be derived using the general result about maximality of so-called -matchings (Theorem 2 of Section 8, page 152 of berge:1973 ), however, here we will give a direct proof that works only for bipartite graphs.

We will require some definitions. For a positive integer , bipartite graph and a -edge-colorable subgraph of define an -augmenting path as follows.

###### Definition 1.

A simple -path is -augmenting, if it is of odd length, the even edges of belong to , the odd edges lie outside and , .

Observe that if contains an -augmenting path , then . In order to see this, consider a subgraph of obtained from by removing the even edges of from and adding the odd edges. Observe that any vertex of has degree at most in , hence is -edge-colorable by Proposition 1. Moreover, .

The following lemma states that the converse is also true.

###### Lemma 2.

Let be a bipartite graph, and let be a -edge-colorable subgraph with . Then contains an -augmenting path.

###### Proof.

For the -edge-colorable subgraph consider all maximum -edge-colorable subgraphs and choose one maximizing . By an alternating component, we will mean a path or an even cycle of whose edges belong to and , alternatively. Observe that any alternating component is either an even cycle or an even path or an odd path. Moreover, since , there is at least one edge in , hence contains at least one alternating component.

We claim that contains no alternating component that is an even cycle. On the opposite assumption, consider a subgraph of obtained from by exchanging the edges on . Observe that the degree of any vertex of is the same as it was in . Hence is -edge-colorable by Proposition 1. Moreover, , hence is maximum -edge-colorable. However , which contradicts our choice of .

Now, consider all alternating components of and among them choose one maximizing . From the previous paragraph we have that is a path. Let us show that is an odd path. Assume that is an even path connecting vertices and . Assume that is incident to an edge of and is incident to on . Let us show that . If , then is incident to an edge . Observe that . If , then either we have an alternating component that is a cycle, or we have an odd cycle. Both of the cases are contradictory. Thus . Now observe that forms an alternating component with more edges than . This contradicts our choice of .

Thus . Consider a subgraph of by exchanging the edges on . Observe that the degree of any vertex of is the same as it was in except which has degree at most and whose degree has decreased by one. Hence is -edge-colorable by Proposition 1. Moreover, , hence is maximum -edge-colorable. However , which contradicts our choice of .

Thus is an odd path. Again let the end-points of be and . If and are incident to edges on , then similarly to previous paragraph, one can show that and . If we exchange the edges of on we would find a larger -edge-colorable subgraph, contradicting the maximality of .

Thus, and are incident to edges on . Similarly to previous paragraph, one can show that and . Now, it is not hard to see that is an -augmenting path. The proof of the lemma is complete. ∎

When is not bipartite, may possess an augmenting path with respect to a maximum -edge-colorable subgraph. Consider the graph from Figure 1, and let be the subgraph colored with and . It is easy to see that is maximum -edge-colorable in , however contains an -augmenting path.

## 3 The main results

In this section, we obtain the main result of the paper. Our first theorem proves a lower bound for in terms of the average of and .

###### Theorem 1.

For any bipartite graph and

 νk(G)≥νk−1(G)+νk+1(G)2.
###### Proof.

Assume that the statement of the theorem is wrong. Let be a counter-example minimizing . We prove a series of claims that establish various properties of .

###### Claim 0.

is connected and .

###### Proof.

If is the graph with one vertex, then clearly it is bipartite and for any , hence it is not a counter-example to our theorem. Thus, . Let us show that is connected. Assume that contains components, which are . We have that for

 νi(G)=νi(G(1))+...+νi(G(t)),

hence

 νk(G) =νk(G(1))+...+νk(G(t))≥νk−1(G(1))+νk+1(G(1))2+...+νk−1(G(t))+νk+1(G(t))2 =νk−1(G)+νk+1(G)2.

Thus, is not a counter-example to our statement contradicting our assumption. Here we used the fact that are smaller than , hence they are not counter-examples to our theorem. The proof of the claim is complete. ∎

###### Claim 0.

For any maximum -edge-colorable subgraph and any maximum -edge-colorable subgraph , we have

 E(Hk−1)∪E(Hk+1)=E(G).
###### Proof.

If for some and , then there exist an edge , such that lies outside and . Hence

 νk−1(G−e)=νk−1(G)

and

 νk+1(G−e)=νk+1(G),

therefore we get:

 νk(G)≥νk(G−e)≥νk−1(G−e)+νk+1(G−e)2=νk−1(G)+νk+1(G)2.

Here we used the fact that the bipartite graph is not a counter-example. ∎

Our next claim states that removing an edge from does not decrease the size of .

###### Claim 0.

For any edge of , we have .

###### Proof.

If (Proposition 4), then

 νk(G) =1+νk(G−e)≥1+νk−1(G−e)+νk+1(G−e)2 =νk−1(G−e)+1+νk+1(G−e)+12≥νk−1(G)+νk+1(G)2.

Here we used the fact that is not a counter-example and Proposition 4 twice. ∎

Our final claim establishes some relations for maximum and minimum degrees of . Its proof makes use of the fan-argument by Vizing stiebitz:2012 ; vizing:1964 .

and .

###### Proof.

Let and be a maximum -edge-colorable and a maximum -edge-colorable subgraphs of , respectively. By Claim 2 is a union of and , hence it is a union of matchings. Thus .

Let us show that . Assume that . If , then is -regular, hence from Proposition 2 we have for . Therefore

 νk(G)=νk−1(G)+νk+1(G)2.

Thus, is not a counter-example. Hence, we can assume that , and therefore . Let be an edge from this set. Then or must be incident to all colors of (apply Lemma 1 with ). Assume that this vertex is . Let us show that is incident to all colors of as well.

On the opposite assumption, assume that misses a color of . Then must be incident to an edge of color in , as . Since and , there is an edge incident to such that . Let the color of in be .

If is missing at , then consider a subgraph of obtained from by removing the edge , adding to and coloring with . Observe that is -edge-colorable, . Hence is maximum -edge-colorable. However, violating Claim 2.

Thus, we can assume that is present at , hence it is different from . Consider the alternating path of starting from . We claim that passes through . If not, we could have exchanged the colors on , remove from , add to , color it with and get a new maximum -edge-colorable subgraph violating Claim 2. Thus, passes through . We claim that it passes first via , then via and . If first passes via , then together with we get an odd cycle contradicting our assumption.

Let be the final part of that starts from . Consider a -edge-colorable subgraph of obtained from as follows: exchange the colors on , color with , color with and remove from . Observe that is -edge-colorable, . Hence is maximum -edge-colorable. However, violating Claim 2.

Thus and must be incident to all colors of , in particular, and . Observe that by Claim 2, any vertex of degree at least must be incident to an edge from . Consider the bipartite graph . Observe that is a -regular bipartite graph with . Hence from Proposition 2, we have for , and therefore

 νk(G)=νk−1(G)+νk+1(G)2.

This means that is not a counter-example to our statement contradicting our assumption. Hence . The proof of the claim is complete. ∎

We are ready to prove the theorem. By Claim 4, , hence there is a vertex with . On the other hand, by Claim 1 is connected and , hence . Thus, there is an edge incident to . By Claim 3, there is a maximum -edge-colorable subgraph that does not contain . By Lemma 1, or for any such . Since , we have for any maximum -edge-colorable subgraph that does not contain .

By Proposition 3, we have . Let us show that . Assume that . Since (Claim 3) and , we have .

Choose a maximum -edge-colorable subgraph of . If is maximum in , then since does not lie in , we have a contradiction with as . Thus is not maximum in . By Lemma 2, there a -edge-colorable subgraph which is obtained from by shifting the edges on an -augmenting path in . Observe that . If is maximum in , then we have a contradiction with as . Thus is not maximum in . By repeating the argument and applying Lemma 2 at most times, we will obtain a maximum -edge-colorable subgraph of with contradicting the fact that for any maximum -edge-colorable subgraph of that does not contain .

Thus, . We have

 νk(G) =k+νk(G−v)≥k+νk−1(G−v)+νk+1(G−v)2 =νk−1(G−v)+k−1+νk+1(G−v)+k+12≥νk−1(G)+νk+1(G)2.

This contradicts the fact that is a counter-example to our statement. Here we used the fact that is not a counter-example and Proposition 3 twice. The proof of the theorem is complete. ∎

The proved theorem is equivalent to the following

###### Remark 1.

If is a bipartite graph, then

 ν1(G)−ν0(G)≥ν2(G)−ν1(G)≥ν3(G)−ν2(G)≥....

Below we derive the main result of the paper as a corollary to the theorem proved above:

###### Corollary 1.

Let be a bipartite graph and let . Then for we have

 νk(G)≥νk−i(G)+νk+i(G)2.
###### Proof.

We prove the statement by induction on . When , the statement is trivial. When , it follows from Theorem 1. We will assume that the statement is true for , and prove it for .

By induction hypothesis we have

 νk(G)≥νk−i+1(G)+νk+i−1(G)2.

By applying Theorem 1 on and we have

 νk(G)≥νk−i+2(G)+νk−i(G)+νk+i−2(G)+νk+i(G)4=νk−i(G)+νk+i(G)4+νk−i+2(G)+νk+i−2(G)4.

So, in order to complete the proof of the corollary, we need to show

 νk−i+2(G)+νk+i−2(G)≥νk−i(G)+νk+i(G). (1)

Using Remark 1, we have

 νk−i+1(G)−νk−i(G)≥νk−i+2(G)−νk−i+1(G)≥⋯≥νk+i−1(G)−νk+i−2(G)≥νk+i(G)−νk+i−1(G).

The last inequality implies

 [νk−i+1(G)−νk−i(G)]+[νk−i+2(G)−νk−i+1(G)]≥[νk+i−1(G)−νk+i−2(G)]+[νk+i(G)−νk+i−1(G)],

or

 νk−i+2(G)−νk−i(G)≥νk+i(G)−νk+i−2(G),

which is equivalent to (1). The proof of the corollary is complete. ∎

## 4 Future Work

For a (not necessarily bipartite) graph , let be the smallest number of vertices of whose removal results into a bipartite graph. One can easily see that coincides with the minimum number of vertices of , such that any odd cycle of contains a vertex from these vertices. is a well studied parameter frequently appearing in various papers on Graph theory and Algorithms. It can be easily seen that a graph is bipartite if and only if , and is nearly bipartite if and only if .

We suspect that:

###### Conjecture 2.

Let be a graph and let . Then for we have

 νk(G)≥νk−i(G)+νk+i(G)−b(G)2.

Observe that when is bipartite, we get the statement of Corollary 1. On the other hand, when is nearly bipartite and , we get the statement of the Conjecture 1.

## Acknowledgement

The second author is indebted to Armen Asratian for useful discussions on -matchings and for disproving an early version of Lemma 2.

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