The threshold for the full perfect matching color profile in a random coloring of random graphs

Consider a graph G with a coloring of its edge set E(G) from a set Q = {c1, c2, . . . , cq}. Let Qi be the set of all edges colored with ci. Recently, Frieze defined a notion of the perfect matching color profile denoted by mcp(G), which is the set of vectors (m1,m2, . . . ,mq) such that there exists a perfect matching M in G with |Qi ∩M | = mi for all i. Let α1, α2, . . . , αq be positive constants such that ∑q i=1 αi = 1. Let G be the random bipartite graph Gn,n,p. Suppose the edges of G are independently colored with color ci with probability αi. We determine the threshold for the event mcp(G) = {(m1, . . . ,mq) ∈ [0, n]q : m1 + · · ·+mq = n}, answering a question posed by Frieze. We further extend our methods to find the threshold for the same event in a randomly colored random graph Gn,p. Mathematics Subject Classifications: 05C80

Throughout this paper, we have the following setting: We are given a graph G, and positive constants α 1 , α 2 , . . . , α q with are independently colored with a random color from the set Q = {c 1 , c 2 , . . . , c q } with probability P (c(e) = c i ) = α i , where c(e) denotes the color of the edge e ∈ E(G). Define the color class Q i = {e ∈ E(G) : c(e) = c i }. The perfect matching color profile mcp(G) is defined to be the set of vectors (m 1 , m 2 , . . . , m q ) such that there exists a perfect matching We first consider G to be the random bipartite graph G n,n,p . For an event E n , we say that E n occurs with high probability (in short, w.h.p.) if P(E n ) → 1 as n → ∞. Erdős and Rényi [3] proved that G n,n,p has a perfect matching w.h.p. when p = log n+ω n for any ω = ω(n) → ∞. Moreover, for the same value of p, Frieze [6] proved that if the edges of G = G n,n,p are independently colored with q colors with constant probabilities, then most of the elements (m 1 , Theorem 1 (Frieze). Let α 1 , α 2 , . . . , α q , β be positive constants such that α 1 + α 2 + · · · + α q = 1 and β < 1/q. Let G be the random bipartite graph G n,n,p where p = log n+ω n , ω = ω(n) → ∞. Suppose that the edges of G are independently colored with colors from . Let m 1 , m 2 , . . . , m q satisfy: (i) m 1 + · · · + m q = n and (ii) m i βn, i ∈ [q]. Then w.h.p., there exists a perfect matching M in which exactly m i edges are colored with c i , i = 1, 2, . . . , q.
It is not hard to check that w.h.p. (n, 0, . . . , 0) / ∈ mcp(G), in view of the fact that the bipartite graph induced by the first color is distributed as G n,n,α 1 p and has isolated vertices w.h.p. Frieze posed the natural problem of determining the threshold for mcp(G) = {(m 1 , . . . , m q ) ∈ [0, n] q : m 1 + · · · + m q = n}. In this paper, we determine that threshold. Theorem 2. Let α 1 , α 2 , . . . , α q be positive constants such that α 1 + α 2 + · · · + α q = 1. Let Let G be the random bipartite graph G n,n,p where p = log n+ω α min n , ω = ω(n) → ∞. Suppose that the edges of G are independently colored with colors from C = {c 1 , c 2 , . . . , c q } where P(c(e) = c i ) = α i for e ∈ E(G), i ∈ [q]. Then, w.h.p. for each m 1 , m 2 , . . . , m q satisfying m 1 +· · ·+m q = n, there exists a perfect matching M in which exactly m i edges are colored with c i , i = 1, 2, . . . , q. In other words, Let us first determine the lower bound on the threshold. Assume that α min = α i . To prove the lower bound, note that it is enough to show that the same threshold holds even for the event that G contains a perfect matching in color c i . To see this, remember that the bipartite graph induced by the color c i is distributed as G n,n,α i p . The claim now follows from the known thresholds of the random bipartite graph to have a perfect matching, see e.g., Theorem 6.1 of [7]. The general strategy to prove the upper bound on the threshold in Theorem 2 is to do the following modification iteratively. For each i = j, if G contains a perfect matching M using m i n q edges with color c i , then we can the electronic journal of combinatorics 28(1) (2021), #P1.21 find a perfect matching M consisting of one fewer edge of color c i and one more edge of color c j . Frieze [6] also suggested studying the same problem for the random graph G n,p . A simple extension of our techniques establishes the threshold for G n,p as well.
Theorem 3. Let α 1 , α 2 , . . . , α q be positive constants such that α 1 + α 2 + · · · + α q = 1. Let Let G be the random graph G n,p where p = log n+ω α min n , ω = ω(n) → ∞. Suppose that the edges of G are independently colored with colors from . Then, w.h.p. for each m 1 , m 2 , . . . , m q satisfying m 1 + · · · + m q = n 2 , there exists a perfect matching M in which exactly m i edges are colored with c i , i = 1, 2, . . . , q. In other words, Similar to Theorem 2, the lower bound on the threshold for Theorem 3 follows from the known thresholds of the random graph to have a perfect matching (see, e.g., Theorem 6.2 of [7]).
This short note is organized as follows. The next section is devoted to stating a few simple structural lemmas about random bipartite graphs and random graphs. Section 3 contains the proof of Theorem 2 and Theorem 3. Finally, we finish with a few concluding remarks.

Structural lemmas
Let α i , 1 i q, and α min be as in Theorems 2 and 3. Throughout this section, the graph G will be either the random bipartite graph G n,n,p or the random graph G n,p , where the probability p = log n+ω α min n , for some ω = ω(n) → ∞. The edges of G are randomly colored as in Theorems 2 and 3. Proof. Note that it is enough to prove this lemma with |X| = |Y | = n 4q . Now by a simple union bound, we have the following: Lemma 5. Let G be the random graph G n,p . Suppose that the edges of G are colored in the exact same way as in Lemma 4. Then, w.h.p. for each i ∈ [q], and any disjoint sets X, Y ⊆ V (G) with |X|, |Y | n 8q , there is an edge with color c i between X and Y in G. Proof. This follows very similarly to the proof of Lemma 4. Lemma 6. Let G be the random bipartite graph G n,n,p or the random graph G n,p . Then, w.h.p. for each i ∈ [q], the graph G contains a perfect matching in color c i .
Proof. This is an easy consequence of Theorems 6.1 and 6.2 of [7].

Proof of the main results
Proof of Theorem 2.
Suppose that we are given a bipartite graph G for which the high probability properties (Lemmas 4 and 6) of the random bipartite graph G n,n,p mentioned in the last section hold. The proof mainly consists of showing that the following can be done. For each i = j, if G contains a perfect matching M with at least n q edges with color c i , then G contains a perfect matching with the same color profile as M but with one fewer edge of color c i and one more edge of color c j . We next show how we can iteratively apply this modification to obtain a perfect matching with any given color profile.
Fix (m 1 , m 2 , . . . , m q ) ∈ [0, n] q such that q i=1 m i = n. Our goal is to show that G has a perfect matching M such that |M ∩ Q i | = m i for all i. Without loss of generality we can assume that m 1 = max {m i : i ∈ [q]}. This implies that m 1 n q . By Lemma 6, we know that there is a perfect matching in the subgraph induced by color c 1 in G. We proceed in the following way: starting with a perfect matching with color profile (n, 0, . . . , 0), for any fixed color c j with j = 1 we show the existence of a perfect matching with one fewer edge in color c 1 and one more edge in color c j . We keep doing this process until we get a matching with m i edges with color c i for all i. Note that we need n − m 1 steps to reach a matching with the color profile (m 1 , m 2 , . . . , m q ), because in every step, we find a matching with one fewer edge in color c 1 . So, it is enough to show that for any perfect matching M in G with |M ∩ Q i | = µ i for each i ∈ [q] and µ 1 n q , there is a matching M in G with |M ∩ Q 1 | = µ 1 − 1, |M ∩ Q 2 | = µ 2 + 1 and |M ∩ Q i | = µ i for all other i.
We show the above statement by finding an appropriate alternating cycle. More precisely, we find a cycle C with vertex sequence (x 1 ∈ A, y 1 ∈ B, x 2 ∈ A, y 2 ∈ B, . . . , x ∈ A, y ∈ B, For the convenience of writing the proof, we introduce some notation. Label vertices so that the edges v Note that if there is an edge with color c 2 between v + i and v − j in G and a directed path from v i to v j in D, then this gives exactly the alternating cycle C which we discussed in the last paragraph. Moreover, by using Lemma 4, we have the following property in D. and If not, then we can pick a minimal set V 1 ⊆ V 1 such that |∪ v∈V 1 B + (v)| n 4q , and note that |∪ v∈V 1 B + (v)| 2n 4q . There are no edges from ∪ v∈V 1 B + (v) into V (D) \ ∪ v∈V 1 B + (v) , and the latter set has size at least |D| − 2n 4q n 4q , contradicting the property (1). Therefore, (1), and therefore there is a directed path from v i to v j in D. This finishes the proof of Theorem 2.
Proof of Theorem 3.
The proof of Theorem 2 extends straightforwardly to a proof of Theorem 3. By Lemma 6, we know that G = G n,p has a perfect matching in each color. Now, if a color profile (m 1 , . . . , m q ) is required (say m 1 is the largest of these), then start with a perfect matching in color c 1 , and split V (G) into A and B arbitrarily so that M is a matching between A and B. The same arguments as in the proof of Theorem 2 can now be used due to Lemma 5, which is the replacement of Lemma 4 we used before. More precisely, to modify a perfect matching M to another matching M with the same color profile but one fewer edge of color c 1 and one more edge of color c j , we choose an arbitrary bipartition V (G) = A ∪ B with M being a matching between A and B, and then implement the exact same argument as before.

Concluding remarks
In this short note, we consider the random bipartite graph G = G n,n,p and the random graph G = G n,p , and determine the threshold on the parameter p for the event that G contains perfect matchings of all color profiles. Some interesting directions of future research would be to determine mcp(G) for Hamilton cycles, spanning trees etc. or to consider deterministic host graphs (e.g., Dirac graphs) instead of random graphs.