A note on randomly colored matchings in random bipartite graphs
Alan Frieze

TL;DR
This paper investigates the structure of perfect matchings in randomly colored bipartite graphs, providing bounds on the distribution of edge colors within such matchings.
Contribution
It introduces the perfect matching color profile and derives bounds for this profile in the context of random bipartite graphs with random edge coloring.
Findings
Bounds on the matching color profile for random bipartite graphs.
Characterization of perfect matchings with specific color distributions.
Insights into the interplay between randomness in graph structure and coloring.
Abstract
We are given a bipartite graph that contains at least one perfect matching and where each edge is colored from a set Q=\{c_1,c_2,\ldots,c_q}\. Let , where denotes the color of . The perfect matching color profile is defined to be the set of vectors such that there exists a perfect matching such that . We give bounds on the matching color profile for a randomly colored random bipartite graph.
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Taxonomy
TopicsLimits and Structures in Graph Theory
A note on randomly colored matchings in random bipartite graphs
Alan Frieze
Department of Mathematical Sciences,
Carnegie Mellon University,
Pittsburgh PA15213,
USA.
[email protected] Research supported in part by NSF grant DMS1661063
Abstract
We are given a bipartite graph that contains at least one perfect matching and where each edge is colored from a set . Let
, where denotes the color of . The perfect matching color profile is defined to be the set of vectors such that there exists a perfect matching such that . We give bounds on the matching color profile for a randomly colored random bipartite graph.
1 Introduction
We consider the following problem: we are given a random bipartite graph in which each edge is given a random color from a set . An edge is colored with probability where is a constant. Let , where denotes the color of . The perfect matching color profile is defined to be the set of vectors such that there exists a perfect matching such that . We give bounds on the matching color profile for a randomly colored random bipartite graph.
Randomly colored random graphs have been studied recently in the context of (i) rainbow matchings and Hamilton cycles, see for example [2], [3], [7], [11]; (ii) rainbow connection see for example [5], [9], [10], [13], [12]; (iii) pattern colored Hamilton cycles, see for example [1], [6]. This paper can be considered to be a contribution in the same genre. One can imagine a possible interest in the color profile via the following scenario: suppose that is a set of tools and is a set of jobs where edge indicates that can be completed using . If colors represent people, then one might be interested in equitably distributing jobs. I.e. determining whether . In any case, we find the problem interesting.
We will consider to be the random bipartite graph where where . Erdős and Rényi [4] proved that has a perfect matching w.h.p. We will prove the following theorem: let be positive constants such that and . Let
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Theorem 1**.**
Let be the random bipartite graph where where . Suppose that the edges of are independently colored with colors from where for . Let satisfy: (i) and (ii) . Then w.h.p., there exists a perfect matching in which exactly edges are colored with .
It is clear that w.h.p. . This is because the bipartite graph induced by edges of color is distributed as and this contains isolated vertices w.h.p. On the other hand, if then w.h.p. . To see this, suppose that . Suppose we have found a matching that uses edges of color for . Let . Then the random bipartite graph induced by vertices not in and having edges of color has density at least and so has a perfect matching w.h.p.
Open Question: What is the threshold for ?
2 Structural Lemma
Suppose that the bipartition of is denoted . For sets we let denote the number of edges of color . We say that vertex is -adjacent to vertex if the edge exists and has color .
Lemma 2**.**
Let where . Then w.h.p.
- (a)
* and and where implies that for .* 2. (b)
There do not exist sets and such that and and such that each is -adjacent to fewer than vertices in . 3. (c)
There do not exist sets and such that and and a set such that each is -adjacent to vetices in . 4. (d)
There do not exist sets and such that such that there are more than vertices in that not -adjacent to a vertex in . 5. (e)
Fix constants. Then w.h.p. there do not exist sets with such that . 6. (f)
There do not exist sets and such that such that .
**Proof
**(a) The probability that the condition is violated can be bounded by
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(b) The probability that the condition is violated can be bounded by
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The factor comes from applying a Chernoff bound.
(c) We can assume w.l.o.g. that . The probability that the condition is violated can be bounded by
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(d) The probability that the condition is violated can be bounded by
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(e) The probability that the condition is violated can be bounded by
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(f) The probability that the condition is violated can be bounded by
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3 Proof of Theorem 1
**Proof **Assume from now on that the high probability conditions of Lemma 2 are in force. Let be a perfect matching and let for . Suppose that and . We show that we can find another matching such that and . We do this by finding an alternating cycle with edge sequence and vertex sequence such that (i) , (ii) , (iii) and (iv) . Repeating this for pairs of colors, one over-subscribed and one under-subscribed we eventually achieve our goal. It is sufficient to consider this case, seeing as we can always w.h.p. find a matching that has been randomly colored with edges of color , .
Next let and for and for let and . Then let
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It follows from Lemma 2(b) that
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It then follows from Lemma 2(c) that if then
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We now define a sequence of sets where is obtained from by adding a vertex of for which . Now consider for some . Then we have
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Given Lemma 2(e) with , we see that this sequence stops with . So we now let . We note that
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We now fix some and define a sequence of sets where and . We let and then having defined we let
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We claim that for ,
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We verify (3) below. Assuming its truth, there exists a smallest such that
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Starting with where , we can similarly construct a sequence of sets where and . Here is the equivalently defined set to in . We can assume that , because of the sizes of the sets . More precisely, by (1), there will be choices for for which . Having defined we let
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and then let . The equivalent of (3) will be
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Assuming its truth, there exists such that
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It follows from Lemma 2(f) that at least 9/10 of the vertices of have a -neighbor in and at least 9/10 of the vertices of have a -neighbor in . We deduce from this that there is a pair such that and . This defines an alternating cycle . Here is a -neighbor of in and is (the reversal of) a path from to and is the path from to , . This completes the proof of Theorem 1.
Verification of (3), (5): We have by the assumption that
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Now suppose that . Then, by (2),
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Applying Lemma 2(a) we see that
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Because the sets expand rapidly, the total size of is small compared with the R.H.S of (7) and (3) follows. The argument for (5) is similar.
4 Concluding Remarks
We have established that w.h.p. is almost all of and posed the question of findng the exact threshold for . It seems technically feasible to extend our results to randomly colored . We leave this for future research. It would be of some interest to analyse other spanning subgraphs from this point of view e.g. Hamilton cycles.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M. Anastos and A.M. Frieze, Pattern Colored Hamilton Cycles in Random Graphs, SIAM Journal on Discrete Mathematics 33 (2019) 528-545.
- 2[2] D. Bal and A.M. Frieze, Rainbow Matchings and Hamilton Cycles in Random Graphs, Random Structures and Algorithms 48 (2016) 503-523.
- 3[3] C. Cooper and A.M. Frieze, Multi-coloured Hamilton cycles in random edge-coloured graphs, Combinatorics, Probability and Computing 11 (2002), 129-134.
- 4[4] P. Erdős and A. Rényi, On random matrices, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1964) 455-461.
- 5[5] A. Dudek, A.M. Frieze and C. Tsourakakis, Rainbow connection of random regular graphs, SIAM Journal on Discrete Mathematics 29 (2015) 2255-2266.
- 6[6] L. Espig, A.M. Frieze and M. Krivelevich, Elegantly colored paths and cycles in edge colored random graphs, SIAM Journal on Discrete Mathematics 32 (2018) 1585-1618.
- 7[7] A. Ferber and M. Krivelevich, Rainbow Hamilton cycles in random graphs and hypergraphs. Recent trends in combinatorics, IMA Volumes in Mathematics and its applications, A. Beveridge, J. R. Griggs, L. Hogben, G. Musiker and P. Tetali, Eds., Springer 2016, 167-189.
- 8[8] A.M. Frieze and P. Loh, Rainbow Hamilton cycles in random graphs, Random Structures and Algorithms 44 (2014) 328-354.
