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 edgecolorable 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 edgecoloring of . Observe that if is incident to the color in the former edgecoloring, 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 edgecolorable, if is edgecolorable and contains maximum number of edges among all edgecolorable subgraphs. For and a graph let
Clearly, a edgecolorable subgraph is maximum if it contains exactly edges.
There are several papers where the ratio has been investigated. Here is a maximum edgecolorable 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 edgecolorable 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 edgecolorable graph in an input graph is NPcomplete 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 edgecolorable 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 graphclasses, 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
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 edgecolorable, 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 edgecolorable graph.
Proposition 3.
If is a graph, is a vertex of and . Then
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
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 edgecolorable subgraph with , we have or .
Proof.
Assume that there is a maximum edgecolorable 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 edgecolorable. 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 edgecolorable 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 endpoints 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 edgecolorable subgraphs of bipartite graphs. It is quite plausible that our result can be derived using the general result about maximality of socalled 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 edgecolorable 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 edgecolorable 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 edgecolorable subgraph with . Then contains an augmenting path.
Proof.
For the edgecolorable subgraph consider all maximum edgecolorable 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 edgecolorable by Proposition 1. Moreover, , hence is maximum edgecolorable. 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 edgecolorable by Proposition 1. Moreover, , hence is maximum edgecolorable. However , which contradicts our choice of .
Thus is an odd path. Again let the endpoints 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 edgecolorable 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 edgecolorable subgraph. Consider the graph from Figure 1, and let be the subgraph colored with and . It is easy to see that is maximum edgecolorable 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
Proof.
Assume that the statement of the theorem is wrong. Let be a counterexample 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 counterexample to our theorem. Thus, . Let us show that is connected. Assume that contains components, which are . We have that for
hence
Thus, is not a counterexample to our statement contradicting our assumption. Here we used the fact that are smaller than , hence they are not counterexamples to our theorem. The proof of the claim is complete. ∎
Claim 0.
For any maximum edgecolorable subgraph and any maximum edgecolorable subgraph , we have
Proof.
If for some and , then there exist an edge , such that lies outside and . Hence
and
therefore we get:
Here we used the fact that the bipartite graph is not a counterexample. ∎
Our next claim states that removing an edge from does not decrease the size of .
Claim 0.
For any edge of , we have .
Proof.
Our final claim establishes some relations for maximum and minimum degrees of . Its proof makes use of the fanargument by Vizing stiebitz:2012 ; vizing:1964 .
Claim 0.
and .
Proof.
Let and be a maximum edgecolorable and a maximum edgecolorable 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
Thus, is not a counterexample. 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 edgecolorable, . Hence is maximum edgecolorable. 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 edgecolorable 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 edgecolorable subgraph of obtained from as follows: exchange the colors on , color with , color with and remove from . Observe that is edgecolorable, . Hence is maximum edgecolorable. 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
This means that is not a counterexample 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 edgecolorable subgraph that does not contain . By Lemma 1, or for any such . Since , we have for any maximum edgecolorable subgraph that does not contain .
Choose a maximum edgecolorable 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 edgecolorable 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 edgecolorable subgraph of with contradicting the fact that for any maximum edgecolorable subgraph of that does not contain .
Thus, . We have
This contradicts the fact that is a counterexample to our statement. Here we used the fact that is not a counterexample 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
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
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
By applying Theorem 1 on and we have
So, in order to complete the proof of the corollary, we need to show
(1) 
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
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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