On maximum k-edge-colorable subgraphs of bipartite graphs

07/17/2018
by   Liana Karapetyan, et al.
Yerevan State University
0

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.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

04/19/2019

On the fixed-parameter tractability of the maximum 2-edge-colorable subgraph problem

A k-edge-coloring of a graph is an assignment of colors {1,...,k} to edg...
07/27/2018

Alternating Path and Coloured Clustering

In the Coloured Clustering problem, we wish to colour vertices of an edg...
05/14/2018

Assembling Omnitigs using Hidden-Order de Bruijn Graphs

De novo DNA assembly is a fundamental task in Bioinformatics, and findin...
01/24/2018

A Chronological Edge-Driven Approach to Temporal Subgraph Isomorphism

Many real world networks are considered temporal networks, in which the ...
03/13/2020

Patch Graph Rewriting

The basic principle of graph rewriting is the stepwise replacement of su...
08/19/2019

On the edge-biclique graph and the iterated edge-biclique operator

A biclique of a graph G is a maximal induced complete bipartite subgraph...
09/04/2019

On Orthogonal Vector Edge Coloring

Given a graph G and a positive integer d, an orthogonal vector d-colorin...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

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

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

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

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 -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.

Figure 1: The statement of Lemma 2 is not true when is not bipartite.

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 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

hence

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

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 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

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 .

Claim 0.

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

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

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

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

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)

Using Remark 1, we have

The last inequality implies

or

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

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.

References

  • [1] M. Albertson and R. Haas. Parsimonious edge coloring. Discrete Mathematics, (148):1–7, 1996.
  • [2] M. Albertson and R. Haas. The edge chromatic difference sequence of a cubic graph. Discrete Mathematics, (177):1–8, 1997.
  • [3] D. Aslanyan, V. Mkrtchyan, S. Petrosyan, and G. Vardanyan. On disjoint matchings in cubic graphs: Maximum 2-edge-colorable and maximum 3-edge-colorable subgraphs. Discrete Applied Mathematics, (172):12–27, 2014.
  • [4] C. Berge. Graphs and Hypergraphs. North Holland Publishing Company, Amsterdam, 1973.
  • [5] B. Bollobas. Extremal Graph Theory. Academic Press, London, New York, San Francisco, 1978.
  • [6] A. Cavicchioli, M. Meschiari, B. Ruini, and F. Spaggiari. A survey on snarks and new results: Products, reducibility and a computer search. Discrete Mathematics, 28(2): 57–86, 1998.
  • [7] A. D. Flaxman and S. Hoory. Maximum matchings in regular graphs of high girth. The Electronic Journal of Combinatorics, 14(1):1–4, 2007.
  • [8] J.-L. Fouquet and J.-M. Vanherpe. On parsimonious edge-colouring of graphs with maximum degree three. Graphs and Combinatorics, 29(3): 475–487, 2013.
  • [9] L. Hambardzumyan and V. Mkrtchyan. Graphs, disjoint matchings and some inequalities. submitted, 2017 (available at https://arxiv.org/pdf/1512.02546.pdf).
  • [10] M. A. Henning and A. Yeo. Tight lower bounds on the size of a maximum matching in a regular graph. Graphs and Combinatorics, 23(6):647–657, 2007.
  • [11] M. J. Kaminski and L. Kowalik. Beyond the Vizing’s bound for at most seven colors. SIAM J. Discrete Math., 28(3):1334–1362, 2014.
  • [12] A. Kosovski. Approximating the maximum 2- and 3-edge-colorable problems. Discrete Applied Mathematics (157): 3593–3600, 2009.
  • [13] V. Mkrtchyan, S. Petrosyan, and G. Vardanyan. On disjoint matchings in cubic graphs. Discrete Mathematics, (310):1588–1613, 2010.
  • [14] V. V. Mkrtchyan and E. Steffen. Maximum -edge-colorable subgraphs of class II graphs. J. Graph Theory, 70 (4), 473–482, 2012.
  • [15] T. Nishizeki. On the maximum matchings of regular multigraphs. Discrete Mathematics, 37:105–114, 1981.
  • [16] T. Nishizeki and I. Baybars. Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Mathematics, 28:255–267, 1979.
  • [17] R. Rizzi. Approximating the maximum 3-edge-colorable subgraph problem. Discrete Mathematics, 309(12):4166–4170, 2009.
  • [18] C. E. Shannon. A theorem on coloring the lines of a network. J. Math. Physics, (28): 148–151, 1949.
  • [19] E. Steffen. Classifications and characterizations of snarks. Discrete Mathematics, (188):183–203, 1998.
  • [20] E. Steffen. Measurements of edge-uncolorability. Discrete Mathematics, (280):191–214, 2004.
  • [21] M. Stiebitz, D. Scheide, B. Toft, and L. M. Favrholdt. Graph Edge Coloring. John Wiley and Sons, 2012.
  • [22] V. Vizing.

    On an estimate of the chromatic class of a

    -graph.
    Diskret Analiz, (3):25–30, 1964.
  • [23] J. Weinstein. Large matchings in graphs. Canadian Journal of Mathematics, 26(6):1498–1508, 1974.
  • [24] D. West. Introduction to Graph Theory. Prentice-Hall, Englewood Cliffs, 1996.