A note on randomly colored matchings in random bipartite graphs

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 $Q_i=\set{e\in E(G):c(e)=c_i}$, where $c(e)$ denotes the color of $e$. The perfect matching color profile $mcp(G)$ is defined to be the set of vectors $(m_1,m_2,\ldots,m_q)\in [n]^q$ such that there exists a perfect matching $M$ such that $|M\cap Q_i|=m_i$. We give bounds on the matching color profile for a randomly colored random bipartite graph.

Equations

$${\alpha }_{\min }=\min \left\{{\alpha }_i:i\in [q]\right\}.$$

$$\begin{gathered} \sum _{s={\gamma }_a\log n}^{n_0}\sum _{t=1}^{{\alpha }_i\eta s\log n}\left(\genfrac{}{}{0ex}{}{n}{s}\right)\left(\genfrac{}{}{0ex}{}{n}{t}\right)\left(\genfrac{}{}{0ex}{}{st}{2{\alpha }_i\eta s\log n}\right)({\alpha }_{i}p{)}^{2{\alpha }_i\eta s\log n} \\ \le \sum _{s={\gamma }_a\log n}^{n_0}\sum _{t=1}^{{\alpha }_i\eta s\log n}{\left(\frac{ne}{s}\right)}^s{\left(\frac{ne}{t}\right)}^t{\left(\frac{est{\alpha }_{i}p}{2{\alpha }_i\eta s\log n}\right)}^{2{\alpha }_i\eta s\log n} \\ \le \sum _{s={\gamma }_a\log n}^{n_0}\sum _{t=1}^{{\alpha }_i\eta s\log n}{\left(\frac{ne}{s}\right)}^s{\left(\frac{ne}{{\alpha }_i\eta s\log n}\right)}^{{\alpha }_i\eta s\log n}{\left(\frac{etp}{2\eta \log n}\right)}^{2{\alpha }_i\eta s\log n} \\ \le \sum _{s={\gamma }_a\log n}^{n_0}\sum _{t=1}^{{\alpha }_i\eta s\log n}{\left({\left(\frac{ne}{s}\right)}^{1/2{\alpha }_i\eta \log n}{\left(\frac{ne}{{\alpha }_i\eta s\log n}\right)}^{1/2}\cdot \frac{e^{1+o(1)}{\alpha }_{i}s\log n}{2n}\right)}^{2{\alpha }_i\eta s\log n} \\ \le \sum _{s={\gamma }_a\log n}^{n_0}\sum _{t=1}^{{\alpha }_i\eta s\log n}{\left(\frac{s\log n}{n}\right)}^{{\alpha }_i\eta s\log n-s}(\log n{)}^s{\left(\frac{e^{3/2+o(1)}{\alpha }_i^{1/2}}{2{\eta }^{1/2}}\right)}^{2{\alpha }_i\eta s\log n}=o(1). \end{gathered}$$

$${\left(\genfrac{}{}{0ex}{}{n}{\beta n}\right)}^2\left(\genfrac{}{}{0ex}{}{\beta n}{{\gamma }_{b}n/\log n}\right){\left(e^{-{\alpha }_i\beta /4}\right)}^{{\gamma }_{b}n}\le {\left({\left(\frac{e}{\beta }\right)}^{(2\beta +o(1))n}{e}^{-{\alpha }_i\beta {\gamma }_b/4}\right)}^n=o(1).$$

$$\begin{gathered} {\left(\genfrac{}{}{0ex}{}{n}{\beta n}\right)}^2\left(\genfrac{}{}{0ex}{}{\beta n}{n/\log n}\right)\left(\genfrac{}{}{0ex}{}{\beta n}{{\gamma }_{b}n/\log n}\right){\left(\left(\genfrac{}{}{0ex}{}{{\gamma }_{b}n/\log n}{k}\right)({\alpha }_{i}p{)}^k\right)}^{n/\log n} \\ \le {\left(\frac{e}{\beta }\right)}^{(2\beta +o(1))n}{\left(\frac{e{\gamma }_b{\alpha }_i}{k}\right)}^{kn/\log n}=o(1). \end{gathered}$$

$${\left(\genfrac{}{}{0ex}{}{n}{\beta n}\right)}^2\left(\genfrac{}{}{0ex}{}{\beta n}{{\gamma }_{d}n/\log n}\right)(1-{\alpha }_{i}p{)}^{\beta n{\gamma }_{d}n/\log n}\le {\left({\left(\frac{e}{\beta }\right)}^{2+o(1)}{e}^{-{\alpha }_i{\gamma }_d}\right)}^{\beta n}=o(1).$$

$$\begin{gathered} {\left(\genfrac{}{}{0ex}{}{n}{\gamma n/\log n}\right)}^2\left(\genfrac{}{}{0ex}{}{{\gamma }^2{n}^2/(\log n{)}^2}{\delta n/\log \log n}\right)p^{\delta n/\log \log n}\le \\ {\left(\frac{e\log n}{\gamma }\right)}^{2\gamma n/\log n}{\left(\frac{{\gamma }^{2}e\log \log n}{\delta \log n}\right)}^{\delta n/\log \log n}=o(1). \end{gathered}$$

$$2^{2n}(1-p{)}^{{\beta }^2{n}^2/100}=o(1).$$

$$D_0^{\prime }$$

$$D_0$$

$$|M(A_2\setminus D_0^{\prime })|\le \frac{{\gamma }_{b}n}{\log n}.$$

$$|W_0|\le \frac{n}{\log n}.$$

$$|S|=|T|\le t+\frac{n}{\log n}\text{ and }{e}_2(S,T)\ge t{k}_0.$$

$$\begin{array}{r}|R_0|\ge \beta n-\frac{5n}{\log n}\\ a\in R_0\text{ implies }|N_2(a)\cap M(R_0)|\ge \frac{{\alpha }_2\beta \log n}{10}-k_0.\end{array}$$

$$Y_i=N_2(X_i)\text{ and }{X}_{i+1}=\left(M^{-1}(Y_i)\setminus \bigcup _{j\le i}{X}_j\right)\cap R_0.$$

$$|X_i|\le \frac{n}{200\log n}\text{ implies that }|X_{i+1}|\ge \frac{{\alpha }_2\beta \log n}{25}|X_i|.$$

$$|X_k|\ge \frac{{\alpha }_2\beta n}{5000}.$$

$${\hat{X}}_i=N_2({\hat{Y}}_i)\text{ and }{\hat{Y}}_{i+1}=\left(M({\hat{X}}_i)\setminus \bigcup _{j\le i}{\hat{Y}}_j\right)\cap {\hat{R}}_0.$$

$$|{\hat{Y}}_i|\le \frac{n}{200\log n}\text{ implies that }|{\hat{Y}}_{i+1}|\ge \frac{{\alpha }_2\beta \log n}{25}|{\hat{Y}}_i|.$$

$$|{\hat{Y}}_{\ell }|\ge \frac{{\alpha }_2\beta n}{5000}.$$

$$|X_1|=|Y_1|\ge \frac{{\alpha }_2\beta \log n}{10}-o(\log n)$$

$$e_2(X_i:(N_2(X_i)\setminus M(A_2\setminus R_0)))\ge \frac{({\alpha }_2\beta \log n)|X_i|}{10+o(1)}.$$

$$|N_2(X_i)\setminus M(A_2\setminus R_0)|\ge \frac{({\alpha }_2\beta \log n)|X_i|}{20+o(1)}.$$

Full text

$Q_{i}=\left\{e\in E(G):c(e)=c_{i}\right\}$, where $c(e)$ denotes the color of $e$. The perfect matching color profile $mcp(G)$ is defined to be the set of vectors $(m_{1},m_{2},\ldots,m_{q})\in[n]^{q}$ such that there exists a perfect matching $M$ such that $|M\cap Q_{i}|=m_{i}$. 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 $G$ in which each edge is given a random color from a set $Q=\left\{c_{1},c_{2},\ldots,c_{q}\right\}$. An edge $e$ is colored $c(e)=c_{i}$ with probability $\alpha_{i}$ where $\alpha_{i}>0$ is a constant. Let $Q_{i}=\left\{e\in E(G):c(e)=c_{i}\right\}$, where $c(e)$ denotes the color of $e$. The perfect matching color profile $mcp(G)$ is defined to be the set of vectors $(m_{1},m_{2},\ldots,m_{q})\in[n]^{q}$ such that there exists a perfect matching $M$ such that $|M\cap Q_{i}|=m_{i}$. 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 $A$ is a set of tools and $B$ is a set of jobs where edge $\left\{a,b\right\}$ indicates that $b$ can be completed using $a$. If colors represent people, then one might be interested in equitably distributing jobs. I.e. determining whether $(n/q,n/q,\ldots,n/q)\in mcp(G)$. In any case, we find the problem interesting.

We will consider $G$ to be the random bipartite graph $G_{n,n,p}$ where $p=\frac{\log n+\omega}{n},\,\omega=\omega(n)\to\infty$ where $\omega=o(\log n)$. Erdős and Rényi [4] proved that $G$ has a perfect matching w.h.p. We will prove the following theorem: let $\alpha_{1},\alpha_{2},\ldots,\alpha_{q},\beta$ be positive constants such that $\alpha_{1}+\alpha_{2}+\cdots+\alpha_{q}=1$ and $\beta<1/q$. Let

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Theorem 1**.**

Let $G$ be the random bipartite graph $G_{n,n,p}$ where $p=\frac{\log n+\omega}{n},\,\omega=\omega(n)\to\infty$ where $\omega=o(\log n)$. Suppose that the edges of $G$ are independently colored with colors from $C=\left\{c_{1},c_{2},\ldots,c_{q}\right\}$ where $mathbb{P}(c(e)=c_{i})=\alpha_{i}$ for $e\in E(G),i\in[q]$. Let $m_{1},m_{2},\ldots,m_{q}$ satisfy: (i) $m_{1}+\cdots+m_{q}=n$ and (ii) $m_{i}\geq\beta n,i\in[q]$. Then w.h.p., there exists a perfect matching $M$ in which exactly $m_{i}$ edges are colored with $c_{i},i=1,2,\dots,q$.

It is clear that w.h.p. $(n,0,\ldots,0)\notin mcp(G)$. This is because the bipartite graph induced by edges of color $c_{1}$ is distributed as $G_{n,n,\alpha_{1}p}$ and this contains isolated vertices w.h.p. On the other hand, if $p\geq\frac{q(\log n+\omega)}{\alpha_{\min}n}$ then w.h.p. $mcp(G)=[n]^{q}$. To see this, suppose that $m_{1}\leq m_{2}\leq\cdots m_{q}\leq n$. Suppose we have found a matching that uses $m_{i}$ edges of color $c_{i}$ for $i\geq 0$. Let $n^{\prime}=n-m_{1}-\cdots-m_{i}$. Then the random bipartite graph induced by vertices not in $M$ and having edges of color $c_{i}$ has density at least $\frac{q\alpha_{i}n^{\prime}}{\alpha_{\min}n}\cdot\frac{\log n+\omega}{n^{\prime}}\geq\frac{\log n^{\prime}+\omega/2}{n^{\prime}}$ and so has a perfect matching w.h.p.

Open Question: What is the threshold for $mcp(G)=[0,n]^{q}$?

2 Structural Lemma

Suppose that the bipartition of $V(G)$ is denoted $A,B$. For sets $S\subseteq A,T\subseteq B$ we let $e_{i}(S,T)$ denote the number of $S:T$ edges of color $c_{i}$. We say that vertex $u$ is $c_{i}$-adjacent to vertex $v$ if the edge $\left\{u,v\right\}$ exists and has color $c_{i}$.

Lemma 2**.**

Let $p=\frac{\log n+\omega}{n},\,\omega=\omega(n)\to\infty$ where $\omega=o(\log n)$. Then w.h.p.

1. (a)

$S\subseteq A,T\subseteq B$* and $\gamma_{a}\log n\leq|S|\leq n_{0}=\gamma_{a}n/\log n$ and $|T|\leq\alpha_{i}\eta|S|\log n$ where $\gamma_{a}=\eta/(20\alpha_{i})$ implies that $e_{i}(S:T)\leq 2\alpha_{i}\eta|S|\log n$ for $i=1,2,\ldots,q$.* 2. (b)

There do not exist sets $X\subseteq S\subseteq A,T\subseteq B$ and $i\in[q]$ such that $|S|,|T|\geq\beta n$ and $|X|=\gamma_{b}|S|/\log n,\gamma_{b}=10\log(e/\beta)/\alpha_{i}$ and such that each $x\in X$ is $c_{i}$-adjacent to fewer than $\alpha_{i}\beta\log n/10$ vertices in $T$. 3. (c)

There do not exist sets $X\subseteq S\subseteq A,T\subseteq B$ and $i\in[q]$ such that $|S|,|T|\geq\beta n$ and $|X|=|S|/\log n$ and a set $Z\subseteq T,|Z|=\gamma_{b}n/\log n$ such that each $x\in X$ is $c_{i}$-adjacent to $k=\frac{10\log n}{\log\log n}$ vetices in $Z$. 4. (d)

There do not exist sets $S\subseteq A,T\subseteq B$ and $i\in[q]$ such that $|S|,|T|\geq\beta n$ such that there are more than $\gamma_{d}n/\log n,\gamma_{d}=\frac{4}{\alpha_{i}}\log\left(\frac{e}{\beta}\right)$ vertices in $T$ that not $c_{i}$-adjacent to a vertex in $S$. 5. (e)

Fix $\gamma,\delta>0$ constants. Then w.h.p. there do not exist sets $S,T$ with $|S|=|T|=\gamma n/\log n$ such that $e_{i}(S,T)\geq\delta|S|\log n/\log\log n$. 6. (f)

There do not exist sets $S\subseteq A,T\subseteq B$ and $i\in[q]$ such that $|S|,|T|\geq\beta n/10$ such that $e_{i}(S,T)=0$.

**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 $e^{-\alpha_{i}\beta\gamma_{b}/4}$ comes from applying a Chernoff bound.

(c) We can assume w.l.o.g. that $|S|=|T|=\beta n$. 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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$\Box$

3 Proof of Theorem 1

**Proof      **Assume from now on that the high probability conditions of Lemma 2 are in force. Let $M$ be a perfect matching and let $\mu_{i}=|M\cap Q_{i}|$ for $i\in[q]$. Suppose that $\mu_{1}>m_{1}\geq\beta n$ and $\beta n\leq\mu_{2}<m_{2}$. We show that we can find another matching $M^{\prime}$ such that $|M^{\prime}\cap Q_{1}|=\mu_{1}-1$ and $|M^{\prime}\cap Q_{2}|=\mu_{2}+1$. We do this by finding an alternating cycle with edge sequence $C=(e_{1},f_{1},\ldots,e_{\ell},f_{\ell})$ and vertex sequence $(x_{1}\in A,y_{1}\in B,x_{2},\ldots,x_{\ell},y_{\ell},x_{1})$ such that (i) $e_{i}=\left\{x_{i},y_{i}\right\}\in M,$, (ii) $f_{i}=\left\{y_{i},x_{i+1}\right\}\notin M,i\in[\ell]$, (iii) $e_{1}\in Q_{1}$ and (iv) $E(C)\setminus\left\{e_{1}\right\}\subseteq Q_{2}$. 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 $\approx\alpha_{i}n$ edges of color $c_{i}$, $i=1,2,\ldots,q$.

Next let $A_{i}=V(M\cap Q_{i})\cap A$ and $B_{i}=V(M\cap Q_{i})\cap B$ for $i\in[q]$ and for $S\subseteq A$ let $N_{i}(S)=\left\{b\in B:\exists a\in S\ s.t.\ .\left\{a,b\right\}\in Q_{i}\right\}$ and $N_{i}(a)=N_{i}(\left\{a\right\})$. 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 $W_{0}=A_{1}\setminus D_{0}$ then

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We now define a sequence of sets $W_{0},W_{1},\ldots$ where $W_{j+1}$ is obtained from $W_{j}$ by adding a vertex of $A_{2}\setminus W_{j}$ for which $|N_{2}(a)\cap M(W_{j})|\geq k_{0}$. Now consider $S=W_{t},T=M(W_{t})$ for some $t\geq 1$. Then we have

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Given Lemma 2(e) with $\delta=5,\gamma=2$, we see that this sequence stops with $t=t^{*}\leq 4n/\log n$. So we now let $R_{0}=A_{2}\setminus W_{t^{*}}$. We note that

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We now fix some $a_{0}\in R_{0}$ and define a sequence of sets $X_{0},Y_{0},X_{1},Y_{1},\ldots$ where $X_{j}\subseteq R_{0}$ and $Y_{j}\subseteq B_{2}$. We let $X_{0}=\left\{a_{0}\right\}$ and then having defined $X_{i},i\geq 0$ we let

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We claim that for $i\geq 0$,

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We verify (3) below. Assuming its truth, there exists a smallest $k$ such that

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Starting with $\widehat{Y}_{0}=\left\{b_{0}\right\}$ where $b_{0}=M(a_{0})\in\hat{R}_{0}$, we can similarly construct a sequence of sets $\widehat{Y}_{1},\widehat{X}_{1},\ldots$ where $\widehat{X}_{j}\subseteq M^{-1}(\hat{R}_{0})$ and $\widehat{Y}_{j}\subseteq\hat{R}_{0}$. Here $\hat{R}_{0}$ is the equivalently defined set to $R_{0}$ in $B_{2}$. We can assume that $b_{0}\in\hat{R}_{0}$, because of the sizes of the sets $R_{0},\hat{R}_{0}$. More precisely, by (1), there will be $o(n)$ choices for $a_{0}$ for which $b_{0}\notin\hat{R}_{0}$. Having defined $\widehat{Y}_{i}$ we let

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and then let $\widehat{Y}_{i+1}=M(\widehat{X}_{i})$. The equivalent of (3) will be

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Assuming its truth, there exists $\ell$ such that

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It follows from Lemma 2(f) that at least 9/10 of the vertices of $A_{1}$ have a $c_{2}$-neighbor in $\hat{T}_{0}$ and at least 9/10 of the vertices of $B_{1}$ have a $c_{2}$-neighbor in $R_{0}$. We deduce from this that there is a pair $x_{0}\in A_{1},y_{0}=M(x_{0})\in B_{1}$ such that $N_{2}(x_{0})\cap\widehat{Y}_{\ell}\neq\emptyset$ and $N_{2}(y_{0})\cap R_{0}\neq\emptyset$. This defines an alternating cycle $x_{0},u_{0},P_{1},b_{0},a_{0},P_{2},v_{0},y_{0},x_{0}$. Here $u_{0}$ is a $c_{2}$-neighbor of $x_{0}$ in $\widehat{Y}_{\ell}$ and $P_{1}$ is (the reversal of) a path from $u_{0}$ to $b_{0}$ and $P_{2}$ is the path from $a_{0}$ to $v_{0}\in X_{k}$, $v_{0}\in N_{2}(y_{0})$. This completes the proof of Theorem 1.

Verification of (3), (5): We have by the assumption $a_{0}\in R_{0}$ that

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Now suppose that $1\leq|X_{i}|\leq n/(200\log n)$. Then, by (2),

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Applying Lemma 2(a) we see that

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Because the sets $X_{1},X_{2},\ldots$ expand rapidly, the total size of $\bigcup_{j\leq i}X_{j}$ is small compared with the R.H.S of (7) and (3) follows. The argument for (5) is similar. $\Box$

4 Concluding Remarks

We have established that w.h.p. $mcp(G)$ is almost all of $[0,n]^{q}$ and posed the question of findng the exact threshold for $mcp(G)=[0,n]^{q}$. It seems technically feasible to extend our results to randomly colored $G_{n,p}$. 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.