A note on large rainbow matchings in edge-coloured graphs

A note on large rainbow matchings in edge-coloured graphs

Abstract

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on vertices with minimum colour degree at least contains a rainbow matching of size at least , provided . In this paper, we show that is sufficient for .

Submitted to Graphs and Combinatorics on April 19, 2012.

1 Introduction

Let be a simple graph, that is, no loops or multiple edges. We write for the vertex set of and for the minimum degree of . An edge-coloured graph is a graph in which each edge is assigned a colour. We say such an edge-coloured is proper if no two adjacent edges have the same colour. A subgraph of is rainbow if all its edges have distinct colours. Rainbow subgraphs are also called totally multicoloured, polychromatic, or heterochromatic subgraphs.

For a vertex of an edge-coloured graph , the colour degree of is the number of distinct colours on the edges incident with . The smallest colour degree of all vertices in is the minimum colour degree of and is denoted by . Note that a properly edge-coloured graph with has .

In this paper, we are interested in rainbow matchings in edge-coloured graphs. The study of rainbow matchings began with a conjecture of Ryser [11], which states that every Latin square of odd order contains a Latin transversal. Equivalently, for odd, every properly -edge-colouring of , the complete bipartite graph with vertices on each part, contains a rainbow copy of perfect matching. In a more general setting, given a graph , we wish to know if an edge-coloured graph contains a rainbow copy of . A survey on rainbow matchings and other rainbow subgraphs in edge-coloured subgraph can be found in [4]. From now onwards, we often refer to for an edge-coloured graph (not necessarily proper) of order .

Li and Wang [9] showed that if , then contains a rainbow matching of size . They further conjectured that if , then contains a rainbow matching of size . This bound is tight for properly edge-coloured complete graphs. LeSaulnier et al. [8] proved that if , then contains a rainbow matching of size . Furthermore, if is properly edge-coloured with or , then there is a rainbow matching of size . The conjecture was later proved in full by Kostochka and Yancey [7].

What happens if we have a larger graph? Wang [12] proved that every properly edge-coloured graph with and contains a rainbow matching of size at least . He then asked if there is a function, , such that every properly edge-coloured graph with and contains a rainbow matching of size . The bound on the size of rainbow matching is sharp, as shown for example by any -edge-coloured -regular graph. If exists, then we trivially have . In fact, for even as there exists Latin square without any Latin transversal (see [1, 13]). Diemunsch et at. [2] gave an affirmative answer to Wang’s question and showed that . The bound was then improved to in [10], and shortly thereafter, to in [3].

Kostochka, Pfender and Yancey [6] considered a similar problem with instead of properly edge-coloured graphs. They showed that if is such that and , then contains a rainbow matching of size . Kostochka [5] then asked: can be improved to a linear bound in ? In this paper, we show that is sufficient for . Furthermore, this implies that for .

Theorem 1.1.

If is an edge-coloured graph on vertices with , then contains a rainbow matching of size , provided for and for .

2 Main Result

We write for . For an edge in , we denote by the colour of and let the set of colours be , the set of natural numbers.

The idea of the proof is as follows. By induction, contains a rainbow matching of size . Suppose that does not contain a rainbow matching of size . We are going to show that there exists another rainbow matching of size in . Clearly, the colours of equal to the colours of . If , then there exists a vertex not in . Since , has a neighbour such that does not use any colour of . Hence, it is easy to deduce that contains a rainbow matching of size .

Proof of Theorem 1.1.

We proceed by induction on . The theorem is trivially true for . So fix and assume that the theorem is true for . Let be an edge-coloured graph with and if and otherwise. Suppose that the theorem is false and so does not contain a rainbow matching of size .

By induction, there exists a rainbow matching in , say with for each . Let be another rainbow matching of size (which could be empty) in vertex-disjoint from . Clearly and the colours on is a subset of , as otherwise contains a rainbow matching of size . Without loss of generality, we may assume that with for each . We further assume that and are chosen such that is maximal. Now, let and . Clearly, if there is an edge in , it must have colour in , otherwise contains a rainbow matching of size , or is not maximal.

Fact A If is an edge in , then .

Furthermore, if is an edge with and , then , otherwise contains a rainbow matching of size . First, we are going to show that . Suppose the contrary, . We then claim the following.

Claim By relabeling the indices of (in the interval ) and swapping the roles of and if necessary, there exist distinct vertices , , …, in such that for the following holds for :

  1. is an edge and .

  2. Let be the vertex set . For any colour , there exists a rainbow matching of size on which does not use any colour in .

  3. Let . If is an edge with , then .

  4. If is an edge with and , then .

  5. If is an edge with and , then or .

Proof of Claim. Let and . Observe that part (d) and (e) of the claim hold for . For each in terms, we are going to find satisfying (a) – (e). Suppose that we have already found .

Note that , so . Let be a vertex in . By the colour degree condition, must incident with at least edges of distinct colours, and in particular, at least distinct coloured edges not using colours in . Then, there exists a vertex such that is an edge with by part (e) of the claim for the case . Without loss of generality, and we set .

Part (b) of the claim is true for colour , simply by taking the edge together with a rainbow matching of size on which does not use any colour in . For colour , we take the edge together with a rainbow matching of size on which does not use any colour in .

To show part (c) of the claim, let be an edge for some . By part (b) of the claim for the case , there exists a rainbow matching of size on which does not use any colour in . Set . Then, is a rainbow matching of size vertex-disjoint from . Now, by considering the pair instead of , we must have . Otherwise, contains a rainbow matching of size or is not maximal.

Let be an edge with , and . Pick a rainbow matching of size on with colours , and a rainbow matching of size on which does not contain any colour in . Then, is a rainbow matching of size in , a contradiction. So for any and , showing part (d) of the claim.

Part (e) of the claim follows easily from Fact A, (c) and (d). This completes the proof of the claim. ∎

Recall that . So we have . Pick a vertex . By part (e) of the claim, adjacent to vertices in or incident with edges of colours in . Hence, has colour degree at most , which contradicts . Thus, we must have as claimed. In summary, we have and with for .

Now, if , then . Pick a vertex and since has colour degree at least , there exists a vertex such that is an edge and . It is easy to see that contains a rainbow matching of size , contradicting our assumption. Therefore, we may assume and .

Since , any vertex must have a neighbour such that . If , then contains a rainbow matching of size . So, without loss of generality, and are edges in with . By symmetry, we may assume that for each , and are edges in with . As , must have a neighbour with . Without loss of generality, we may assume for some and . Now, is a rainbow matching of size in , which again is a contradiction. This completes the proof of the theorem. ∎

3 Remarks

In Theorem 1.1, the bound on , the number of vertices, is sharp for (and trivially for ), as shown by properly 3-edge-coloured for and by properly 3-edge-coloured two disjoint copies of for . However, we do not know if the bound is sharp for .

Question.

Given , what is the minimum such that every edge-coloured graph of order with contains a rainbow matching of size ?

Acknowledgment

The authors thank Alexandr Kostochka for suggesting the problem during ‘Probabilistic Methods in Graph Theory’ workshop at University of Birmingham, United Kingdom. We would also like to thank Daniela Kühn, Richard Mycroft and Deryk Osthus for organizing this nice event.

References

  1. R. A. Brualdi and H. J. Ryser, Combinatorics matrix theory, Encyclopedia of Mathematics and its Applications, vol. 39, Cambridge University Press, 1991.
  2. J. Diemunsch, M. Ferrara, C. Moffatt, F. Pfender, and P. S. Wenger, Rainbow matching of size in properly-colored graphs, arXiv:1108.2521.
  3. J. Diemunsch, M. Ferrara, A. Lo, C. Moffatt, F. Pfender, and P. S. Wenger, Rainbow matching of size in properly-colored graphs, submitted.
  4. M. Kano and X. Li, Monochromatic and heterochromatic subgraphs in edge-colored graphs – a survey, Graphs Combin. 24 (2008), 237-263.
  5. A. Kostochka, personal communication.
  6. A. Kostochka and M. Yancey, Large rainbow matchings in edge-coloured graphs, Combinatorics, Probability and Computing 21 (2012), 255-263.
  7. A. Kostochka , F. Pfender and M. Yancey, Large rainbow matchings in large graphs, arXiv:1204.3193.
  8. T. D. LeSaulnier, C. Stocker, P. S.Wenger, and D. B. West, Rainbow matching in edge-colored graphs, Electron. J. Combin. 17 (2010), #N26.
  9. H. Li and G. Wang, Heterochromatic matchings in edge-colored graphs, Electron. J. Combin. 15 (2008), #R138.
  10. A. Lo, A note on rainbow matchings in properly edge-coloured graphs, arXiv:1108.5273.
  11. H. J. Ryser, Neuere probleme der kombinatorik, Vortrage über Kombinatorik Oberwolfach, Mathematisches Forschungsinstitut Oberwolfach (1967), 24-29.
  12. G. Wang, Rainbow matchings in properly edge colored graphs, Electron. J. Combin. 18 (2011), #P162.
  13. I. M. Wanless, Transversals in latin squares: a survey, Surveys in Combinatorics 2011, London Math. Soc., 2011.
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