Rainbow Matchings of size δ(G) in Properly Edge-colored Graphs
Jennifer Diemunsch, Michael Ferrara, Casey Moffatt, Florian Pfender, Paul S. Wenger
aa r X i v : . [ m a t h . C O ] A ug Rainbow Matchings of Size δ ( G ) in ProperlyEdge-Colored Graphs Jennifer Diemunsch ∗‡ Michael Ferrara ∗§ , Casey Moffatt ∗ ,Florian Pfender † , and Paul S. Wenger ∗ August 30, 2018
Abstract A rainbow matching in an edge-colored graph is a matching in which all the edgeshave distinct colors. Wang asked if there is a function f ( δ ) such that a properly edge-colored graph G with minimum degree δ and order at least f ( δ ) must have a rainbowmatching of size δ . We answer this question in the affirmative; f ( δ ) = 6 . δ suffices.Furthermore, the proof provides a O ( δ ( G ) | V ( G ) | )-time algorithm that generates sucha matching. Keywords:
Rainbow matching, properly edge-colored graphs
All graphs under consideration in this paper are simple, and we let δ ( G ) and ∆( G ) denote theminimum and maximum degree of a graph G , respectively. A rainbow subgraph in an edge-colored graph is a subgraph in which all edges have distinct colors. Rainbow matchings are ofparticular interest given their connection to transversals of Latin squares: each Latin squarecan be converted to a properly edge-colored complete bipartite graph, and a transversal ofthe Latin square is a perfect rainbow matching in the graph. Ryser’s conjecture [2] thatevery Latin square of odd order has a transversal can be seen as the beginning of the studyof rainbow matchings. Stein [5] later conjectured that every Latin square of order n hasa transversal of size n −
1; equivalently every properly edge-colored K n,n has a rainbow ∗ Dept. of Mathematical and Statistical Sciences, Univ. of Colorado Denver, Denver, CO; email addresses [email protected] , [email protected] , [email protected] , [email protected] . † Institut f¨ur Mathematik, Univ. Rostock, Rostock, Germany;
[email protected] . ‡ Research supported in part by UCD GK12 Transforming Experiences Project, NSF award § Research supported in part by Simons Foundation Collaboration Grant n −
1. The connection between Latin transversals and rainbow matchingsin K n,n has inspired additional interest in the study of rainbow matchings in triangle-freegraphs.Several results have been attained for rainbow matchings in arbitrarily edge-coloredgraphs. The color degree of a vertex v in an edge-colored graph G , written ˆ d ( v ), is thenumber of different colors on edges incident to v . We let ˆ δ ( G ) denote the minimum colordegree among the vertices in G . Wang and Li [7] proved that every edge-colored graph G contains a rainbow matching of size at least l δ ( G ) − m , and conjectured that l ˆ δ ( G ) / m couldbe guaranteed when ˆ δ ( G ) ≥
4. LeSaulnier et al. [4] then proved that every edge-coloredgraph G contains a rainbow matching of size j ˆ δ ( G ) / k . Finally, Kostochka and Yancey [3]proved the conjecture of Wang and Li in full, and also that triangle-free graphs have rainbowmatchings of size l δ ( G )3 m .Since the edge-colored graphs generated by Latin squares are properly edge-colored, it isof interest to consider rainbow matchings in properly edge-colored graphs. In this direction,LeSaulnier et al. proved that a properly edge-colored graph G satisfying | V ( G ) | 6 = δ ( G ) + 2that is not K has a rainbow matching of size ⌈ δ ( G ) / ⌉ . Wang then asked if there is afunction f such that a properly edge-colored graph G with minimum degree δ and order atleast f ( δ ) must contain a rainbow matching of size δ [6]. As a first step towards answeringthis question, Wang showed that a graph G with order at least δ has a rainbow matchingof size j δ ( G )5 k .In this paper we answer Wang’s question from [6] in the affirmative. Theorem 1. If G is a properly edge-colored graph satisfying | V ( G ) | > δ − + δ , then G contains a rainbow matching of size δ ( G ) . If G is triangle-free, a smaller order suffices. Theorem 2. If G is a triangle-free properly edge-colored graph satisfying | V ( G ) | > δ − + δ , then G contains a rainbow matching of size δ ( G ) . The proofs of Theorems 1 and 2 depend on the implementation of a greedy algorithm,a significantly different approach than those found in [3], [4], [6], and [7]. This algorithmgenerates a rainbow matching in a properly edge-colored graph G in O ( δ ( G ) | V ( G ) | )-time.Since there are n × n Latin squares with no transversals (see [1]) when n is even, it isclear that f ( δ ) > δ when δ is even. Furthermore, since maximum matchings in K δ,n − δ haveonly δ edges (provided n ≥ δ ), there is no function for the order of G depending on δ ( G )that can guarantee a rainbow matching of size greater than δ .2 Proof of the Main Results
Proof of Theorem 1.
We proceed by induction on δ ( G ). The result is trivial if δ ( G ) = 1. Weassume that G is a graph with minimum degree δ and order greater than δ − + δ . Lemma 3. If G satisfies ∆( G ) > δ − , then G has a rainbow matching of size δ .Proof. Let v be a vertex of maximum degree in G . By induction, there is a rainbow matching M of size δ − G − v . Since v is incident to at least 3 δ − v that is not incident to any edge in M and also has a color thatdoes not appear in M . Thus there is a rainbow matching of size δ in G . Lemma 4. If G has a color class containing at least δ − edges, then G has a rainbowmatching of size δ .Proof. Let C be a color class with at least 2 δ − M of size δ − G − C . There are 2 δ − M , thusone of the edges in C has no endpoint covered by M , and the matching can be extended.The proof of Theorem 1 relies on the implementation of a greedy algorithm. We beginby preprocessing the graph so that each edge is incident to at least one vertex with degree δ . To achieve this, we order the edges in G and process them in order. If both endpoints ofan edge have degree greater than δ when it is processed, delete that edge. In the resultinggraph, every edge is incident to a vertex with degree δ . Furthermore, by Lemma 3 we mayassume that ∆( G ) ≤ δ −
3; thus the degree sum of the endpoints of any edge is boundedabove by 4 δ −
3. After preprocessing, we begin the greedy algorithm.In the i th step of the algorithm, a smallest color class is chosen (without loss of generality,color i ), and then an edge e i of color i is chosen such that the degree sum of the endpointsis minimum. All the remaining edges of color i and all edges incident with an endpoint of e i are deleted. The algorithm terminates when there are no edges in the graph.We assume that the algorithm fails to produce a matching of size δ in G ; suppose thatthe rainbow matching M generated by the algorithm has size k . We let R denote the set ofvertices that are not covered by M .Let c i denote the size of the smallest color class at step i . Since at most two edges ofcolor i + 1 are deleted in step i (one at each endpoint of e i ), we observe that c i +1 + 2 ≥ c i .Otherwise, at step i color class i + 1 has fewer edges. Let step h be the last step in thealgorithm in which a color class that does not appear in M is completely removed from G .3t then follows that c h ≤
2, and in general c i ≤ h − i + 1) for i ∈ [ h ]. Let f i denote thenumber of edges of color i deleted in step i with both endpoints in R . Since f i < c i , we have f i ≤ h − i ) + 1 for i ∈ [ h ]. Note that after step h , there are exactly k − h colors remainingin G . By Lemma 4, color classes contain at most 2 δ − k − h steps remove at most ( k − h )(2 δ −
2) edges. Furthermore, for i > h , the degree sum of theendpoints of e i is at most 2( δ − i ∈ [ h ], let x i and y i be the endpoints of e i , and let d i ( v ) denote the degree ofa vertex v at the beginning of step i . Let µ i = max { , d i ( x i ) + d i ( y i ) − δ } ; note that2 δ ≤ δ + µ i ≤ δ −
3. Thus, at step i , at most 2 δ + µ i + f i − | E ( G ) | ≤ ( k − h )(2 δ −
2) + h X i =1 (2 δ + µ i + f i − . (1)We now compute a lower bound for the number of edges in G . Since the degree sum ofthe endpoints of e i in G is at least 2 δ + µ i , we immediately obtain the following inequality: nδ + P i ∈ [ h ] µ i ≤ | E ( G ) | . If f i > µ i >
0, then there is an edge with color i having both endpoints in R . Sincethis edge was not chosen in step i by the algorithm, the degree sum of its endpoints is atleast 2 δ + µ i , and one of its endpoints has degree at least δ + µ i . For each value of i satisfying f i >
0, we wish to choose a representative vertex in R with degree at least δ + µ i . Since thereare f i edges with color i with both endpoints in R , there are f i possible representatives forcolor i . Since a vertex in R with high degree may be the representative for multiple colors,we wish to select the largest system of distinct representatives.Suppose that the largest system of distinct representatives has size t , and let T be theset of indices of the colors that have representatives. For each color i ∈ T there is a distinctvertex in R with degree at least δ + µ i . Thus we may increase the edge count of G as follows: nδ + P i ∈ [ h ] µ i + P i ∈ T µ i ≤ | E ( G ) | . (2)We let { f ↓ i } denote the sequence { f i } i ∈ [ h ] sorted in nonincreasing order. Since f i ≤ h − i )+1, we conclude that f ↓ i ≤ h − i )+1. Because there is no system of distinct representativesof size t + 1, the sequence { f ↓ i } cannot majorize the sequence { t + 1 , t, t − , . . . , } . Hencethere is a smallest value p ∈ [ t + 1] such that f ↓ p ≤ t + 1 − p . Therefore, the maximum value4f P hi =1 f ↓ i is bounded by the sum of the sequence { h − , h − , . . . , h − p ) + 3 , t + 1 − p, . . . , t + 1 − p } . Summing we attain X i ∈ [ h ] f i ≤ ( p − h − p + 1) + ( h − p + 1)( t + 1 − p ) . Over p , this value is maximized when p = t +1, yielding P i ∈ [ h ] f i ≤ t (2 h − t ). Since h ≤ δ − P i ∈ [ h ] f i ≤ δ − t − t .We now combine bounds (1) and (2): nδ + P i ∈ [ h ] µ i + P i ∈ T µ i ≤ ( k − h )(2 δ −
2) + h X i =1 (2 δ + µ i + f i − . Hence, since k ≤ δ − nδ ≤ (2 δ − δ −
1) + 12 X [ h ] \ T µ i + X i ∈ [ h ] f i ≤ (2 δ − δ −
1) + ( δ − − t )( δ − /
2) + 2( δ − t − t ≤ δ − δ + 52 + (cid:18) δ − (cid:19) t − t . This bound is maximized when t = ( δ − ) /
2. Thus n ≤ δ −
232 + 418 δ , contradicting our choice for the order of G . Sketch of Proof of Theorem 2.
When G is triangle-free, Lemma 3 can be improved. In par-ticular, ∆( G ) ≤ δ − δ −
1. Since ∆( G ) is used to bound the value of µ i in theproof of Theorem 1, the same argument yields the following inequality: nδ ≤ (2 δ − δ −
1) + 12 X [ h ] \ T µ i + X i ∈ [ h ] f i ≤ (2 δ − δ −
1) + 12 ( δ − − t )( δ −
2) + 2( δ − t − t ≤ δ − δ + 2 + (cid:18) δ − (cid:19) t − t . This upper bound is maximized when t = ( δ − /
2, yielding n ≤ δ −
212 + 92 δ . Conclusion
The proof of Theorem 1 provides the framework of a O ( δ ( G ) | V ( G ) | )-time algorithm thatgenerates a rainbow matching of size δ ( G ) in a properly edge-colored graph G . Given such a G , we create a sequence of graphs { G i } as follows, letting G = G , δ = δ ( G ), and n = | V ( G ) | .First, determine δ ( G i ), ∆( G i ), and the maximum size of a color class in G i ; this process takes O ( n )-time. If ∆( G i ) ≤ δ ( G i ) − δ ( G i ) − G i = G ′ . If ∆( G i ) > δ ( G i ) −
3, then delete a vertex v of maximum degree and then process the edges of G i − v , iteratively deleting those withtwo endpoints of degree at least δ ( G i ); the resulting graph is G i +1 . If ∆( G i ) ≤ δ ( G i ) − C has at least 2 δ ( G i ) − C and then processthe edges of G i − C , iteratively deleting those with two endpoints of degree at least δ ( G i );the resulting graph is G i +1 . Note that δ ( G i +1 ) = δ ( G i ) −
1. If this process generates G δ , weset G ′ = G δ and terminate. Generating the sequence { G i } consists of at most δ steps, eachtaking O ( n )-time.Given that G ′ = G i , the algorithm from the proof of Theorem 1 takes O ( δn )-time togenerate a matching of size δ − i in G ′ . The preprocessing step and the process of determininga smallest color class and choosing an edge in that class whose endpoints have minimumdegree sum both take O ( n )-time. This process is repeated at most δ times.A matching of size δ − ( i + 1) in G i +1 is easily extended in G i to a matching of size δ − i using the vertex of maximum degree or maximum color class. The process of extending thematching takes O ( δ )-time. Thus the total run-time of the algorithm generating the rainbowmatching of size δ in G is O ( δn ).It is worth noting that the analysis of the greedy algorithm used in the proof of Theorem 1could be improved. In particular, the bound c i +1 ≥ c i − i there arean equal number of edges of color i and i + 1 and both endpoints of e i are incident to edgeswith color i + 1. However, since one of the endpoints of e i has degree at most δ , at most δ − i . Since the maximum size of a color class in G is at most 2 δ −
2, if G has order at least 6 δ , then there are at least 3 δ/ i , the bound c i ≤ k − i + 1) can likely be improved. However, we doubtthat such analysis of this algorithm can be improved to yield a bound on | V ( G ) | better than6 δ . 6 eferences [1] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge UniversityPress, Cambridge, UK, 1991.[2] H. J. Ryser, Neuere Probleme der Kombinatorik, in “Vortr¨age ¨uber Kombinatorik Ober-wolfach”. Mathematisches Forschungsinstitut Oberwolfach, July 1967, 24-29.[3] A. Kostochka and M. Yancey, Large Rainbow Matchings in Edge-Colored Graphs. InPreparation.[4] T. D. LeSaulnier, C. Stocker, P. S. Wenger, and D. B. West, Rainbow matching in edge-colored graphs. Electron. J. Combin.
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