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Given an undirected graph $G=(V,E)$, use mathematical induction to show how to construct a graph with $2n$ nodes and $n^2$ edges such that the graph has exactly one, unique, complete pairing.

Note: A pairing is a set $P⊆E$ of edges such that for all $(u,v),(x,y)∈P$, the nodes $u, v, x, y$ are all different. In other words, no two edges in $P$ have a node in common. A complete pairing is a pairing P that uses all the graph’s nodes, that is, a pairing for which

$\bigcup_{(u,v)∈P}${$u,v$}$=V $.

I'm struggling a lot with how to approach this problem. I do understand what a pairing is and I can think of some example graphs with $2n$ nodes and $n^2$ edges s.t. there is a complete pairing (I've drawn a few) but I'm unsure how to approach it in terms of mathematical induction. I can't seem to figure out a base case without drawing a graph or how to approach it mathematically. Any ideas? I would really appreciate any help, hints or a start.

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Look at your graphs for $n=1,n=2,n=3$. The fact that we are supposed to use induction suggests that the $n=1$ graph is a subgraph of the $n=2$ graph and the $n=2$ graph is a subgraph of the $n=3$ graph. Try to draw the graphs in a way that it is clear what is added as you go up in $n$.

When we go from $n$ to $n+1$ we have to add two vertices and $2n+1$ edges. I drew the vertices in two columns, with the top row being the $n=1$ vertices with an edge between them. Now put another pair below those and draw the $3$ edges you add to make the $n=2$ graph in another color. Put a third pair below those and draw the five edges you add to make the $n=3$ case in a third color.

You should be able to see a pattern in how you are adding edges. Your task in the induction is to describe that pattern and show that it can continue as long as you want.

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  • $\begingroup$ Sorry, it was supposed to be $n^2$ edges $\endgroup$ – Lstan14 Apr 14 '19 at 16:16
  • $\begingroup$ Nvm, that makes a lot of sense. Thank you! $\endgroup$ – Lstan14 Apr 14 '19 at 16:18
  • $\begingroup$ Yes I saw there were $n^2$ edges, which is why going from $n^2$ to $(n+1)^2$ you add $2n+1$ edges $\endgroup$ – Ross Millikan Apr 14 '19 at 16:53

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