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In a class each boy knows precisely $d$ girls and each girl knows precisely $d $ boys. Use a result on edge-colouring to show that the boys and girls can be paired off in friendly pairs in at least $d$ different ways.

Please help :)

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A theorem: In this MSE question, it is shown that a bipartite graph $G$ has an edge coloring with $\Delta(G)$ colors, where $\Delta(G)$ is the maximal degree of $G$.

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  • $\begingroup$ Ok, but how to define problem in bipartite graph? And why $\Delta (G) $ is number of DIFFERENT way? $\endgroup$ – user180834 Dec 8 '14 at 21:43
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    $\begingroup$ You have boys and girls, and some boys and girls know each other (I understand that if a boy $x$ knows a girl $y$, then $y$ also knows $x$, am I right?). How would you express this as a bipartite graph? $\endgroup$ – JiK Dec 8 '14 at 21:46
  • $\begingroup$ "I understand that if a boy x knows a girl y, then y also knows x, am I right?" I understand it the same. And my interpretation: i.imgur.com/Zs6Fm2m.jpg @JiK, please glance: math.stackexchange.com/questions/1057889/colouring-graphs-edges $\endgroup$ – user180834 Dec 8 '14 at 21:52
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    $\begingroup$ Now, what does the theorem say about the bipartite graph in your link? $\endgroup$ – JiK Dec 8 '14 at 21:54
  • $\begingroup$ Thanks, now everything become clear. $\endgroup$ – user180834 Dec 8 '14 at 22:13
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Hint:

  • If you put boys on one side and girls on the other you will get a bipartite $d$-regular graph.
  • By counting the edges you will get that the number of boys and girls is the same.
  • Using the Hall's theorem you will get a perfect matching.
  • Remove the above matching from the graph to get a $(d-1)$-regular graph.
  • Repeat until you have $d$ disjoint matchings, color each using a different color.
  • Use induction to formalize the above.
  • There is related question here.

I hope this helps $\ddot\smile$

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  • $\begingroup$ dtldarek@ Are you from Poland :)? $\endgroup$ – user180834 Dec 8 '14 at 22:08
  • $\begingroup$ @user180834 Indeed I am, how do you know? $\endgroup$ – dtldarek Dec 8 '14 at 22:14
  • $\begingroup$ I am also and your name is dtlDAREK :) $\endgroup$ – user180834 Dec 8 '14 at 22:17
  • $\begingroup$ @Darek, could you help here? :)math.stackexchange.com/questions/1057889/colouring-graphs-edges $\endgroup$ – user180834 Dec 8 '14 at 22:27
  • $\begingroup$ Well, you are very perceptive, graph theory should't be any problem, good luck to you :-) $\endgroup$ – dtldarek Dec 8 '14 at 22:28

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