Mixing
Prove that there exists a set $Y$ such that for every $v$, there exists $y in Y$ that is incident to $v$.
$begingroup$
Suppose $A,B,X$ are independent and disjoint sets of vertices in a graph such that $A cup B cup X = V$, $|A|=|B|=9$ and $|X| = 63$. Also, assume $d(v) = 7$ for every $v in A,B$ and suppose that for every $x in X$, $x$ is incident to exactly one $a in A$ and one $b in B$. Prove that there exists a set $Y subseteq X$ such that for every $v in A cup B$, there exists $y in Y$ such that $y$ is incident to $v$, and $|Y|=9$.
I tried to solve using the pigeonhole principle but the numbers didn't work. Any ideas?
combinatorics discrete-mathematics graph-theory pigeonhole-principle matching-theory
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
$endgroup$
add a comment |
$begingroup$
Suppose $A,B,X$ are independent and disjoint sets of vertices in a graph such that $A cup B cup X = V$, $|A|=|B|=9$ and $|X| = 63$. Also, assume $d(v) = 7$ for every $v in A,B$ and suppose that for every $x in X$, $x$ is incident to exactly one $a in A$ and one $b in B$. Prove that there exists a set $Y subseteq X$ such that for every $v in A cup B$, there exists $y in Y$ such that $y$ is incident to $v$, and $|Y|=9$.
I tried to solve using the pigeonhole principle but the numbers didn't work. Any ideas?
combinatorics discrete-mathematics graph-theory pigeonhole-principle matching-theory
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
$endgroup$
add a comment |
$begingroup$
Suppose $A,B,X$ are independent and disjoint sets of vertices in a graph such that $A cup B cup X = V$, $|A|=|B|=9$ and $|X| = 63$. Also, assume $d(v) = 7$ for every $v in A,B$ and suppose that for every $x in X$, $x$ is incident to exactly one $a in A$ and one $b in B$. Prove that there exists a set $Y subseteq X$ such that for every $v in A cup B$, there exists $y in Y$ such that $y$ is incident to $v$, and $|Y|=9$.
I tried to solve using the pigeonhole principle but the numbers didn't work. Any ideas?
combinatorics discrete-mathematics graph-theory pigeonhole-principle matching-theory
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
$endgroup$
Suppose $A,B,X$ are independent and disjoint sets of vertices in a graph such that $A cup B cup X = V$, $|A|=|B|=9$ and $|X| = 63$. Also, assume $d(v) = 7$ for every $v in A,B$ and suppose that for every $x in X$, $x$ is incident to exactly one $a in A$ and one $b in B$. Prove that there exists a set $Y subseteq X$ such that for every $v in A cup B$, there exists $y in Y$ such that $y$ is incident to $v$, and $|Y|=9$.
I tried to solve using the pigeonhole principle but the numbers didn't work. Any ideas?
combinatorics discrete-mathematics graph-theory pigeonhole-principle matching-theory
combinatorics discrete-mathematics graph-theory pigeonhole-principle matching-theory
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
edited Feb 4 at 20:35
Alex Ravsky
42.7k32383
42.7k32383
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
asked Jan 27 at 16:33
Amit LevyAmit Levy
747
747
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
Let $G$ be the given graph and $G’$ be an auxiliary bipartite graph with vertex parts $A$ and $B$ where vertices $ain A$ and $bin B$ are adjacent iff there exists a vertex $xin X$ adjacent in $G$ to both $a$ and $b$. We claim that $G'$ has a perfect matching. To show this we have to check that $G'$ satisfies the conditions of Hall’s Marriage Theorem. Let $W$ be a subset of $A$. Then in $G$ there are $7|W|$ edges incident to vertices of $W$. Each of these edges has a unique adjacent edge of a form $(x,b)$ with some $xin X$ and $bin B$. Then clearly $bin N_{G’}(A)$. Since any vertex $bin B$ can participate in at most $7$ pairs $(x,b)$ for some $xin X$,
$|N_{G’}(A)|ge 7|W|/7=|W|$. Let $M$ be a perfect matching for $G’$. Then $M$ consists of $|A|=|B|=9$ edges and for each edge $(a,x)in M$ there exists a vertex $v(e)in X$ such that $(a,x)$ and $(x,b)$ are edges of the graph $G$. It remains to put $Y={ v(e):ein M}$.
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
$endgroup$
add a comment |
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$begingroup$
Let $G$ be the given graph and $G’$ be an auxiliary bipartite graph with vertex parts $A$ and $B$ where vertices $ain A$ and $bin B$ are adjacent iff there exists a vertex $xin X$ adjacent in $G$ to both $a$ and $b$. We claim that $G'$ has a perfect matching. To show this we have to check that $G'$ satisfies the conditions of Hall’s Marriage Theorem. Let $W$ be a subset of $A$. Then in $G$ there are $7|W|$ edges incident to vertices of $W$. Each of these edges has a unique adjacent edge of a form $(x,b)$ with some $xin X$ and $bin B$. Then clearly $bin N_{G’}(A)$. Since any vertex $bin B$ can participate in at most $7$ pairs $(x,b)$ for some $xin X$,
$|N_{G’}(A)|ge 7|W|/7=|W|$. Let $M$ be a perfect matching for $G’$. Then $M$ consists of $|A|=|B|=9$ edges and for each edge $(a,x)in M$ there exists a vertex $v(e)in X$ such that $(a,x)$ and $(x,b)$ are edges of the graph $G$. It remains to put $Y={ v(e):ein M}$.
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
$endgroup$
add a comment |
$begingroup$
Let $G$ be the given graph and $G’$ be an auxiliary bipartite graph with vertex parts $A$ and $B$ where vertices $ain A$ and $bin B$ are adjacent iff there exists a vertex $xin X$ adjacent in $G$ to both $a$ and $b$. We claim that $G'$ has a perfect matching. To show this we have to check that $G'$ satisfies the conditions of Hall’s Marriage Theorem. Let $W$ be a subset of $A$. Then in $G$ there are $7|W|$ edges incident to vertices of $W$. Each of these edges has a unique adjacent edge of a form $(x,b)$ with some $xin X$ and $bin B$. Then clearly $bin N_{G’}(A)$. Since any vertex $bin B$ can participate in at most $7$ pairs $(x,b)$ for some $xin X$,
$|N_{G’}(A)|ge 7|W|/7=|W|$. Let $M$ be a perfect matching for $G’$. Then $M$ consists of $|A|=|B|=9$ edges and for each edge $(a,x)in M$ there exists a vertex $v(e)in X$ such that $(a,x)$ and $(x,b)$ are edges of the graph $G$. It remains to put $Y={ v(e):ein M}$.
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
$endgroup$
add a comment |
$begingroup$
Let $G$ be the given graph and $G’$ be an auxiliary bipartite graph with vertex parts $A$ and $B$ where vertices $ain A$ and $bin B$ are adjacent iff there exists a vertex $xin X$ adjacent in $G$ to both $a$ and $b$. We claim that $G'$ has a perfect matching. To show this we have to check that $G'$ satisfies the conditions of Hall’s Marriage Theorem. Let $W$ be a subset of $A$. Then in $G$ there are $7|W|$ edges incident to vertices of $W$. Each of these edges has a unique adjacent edge of a form $(x,b)$ with some $xin X$ and $bin B$. Then clearly $bin N_{G’}(A)$. Since any vertex $bin B$ can participate in at most $7$ pairs $(x,b)$ for some $xin X$,
$|N_{G’}(A)|ge 7|W|/7=|W|$. Let $M$ be a perfect matching for $G’$. Then $M$ consists of $|A|=|B|=9$ edges and for each edge $(a,x)in M$ there exists a vertex $v(e)in X$ such that $(a,x)$ and $(x,b)$ are edges of the graph $G$. It remains to put $Y={ v(e):ein M}$.
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
$endgroup$
Let $G$ be the given graph and $G’$ be an auxiliary bipartite graph with vertex parts $A$ and $B$ where vertices $ain A$ and $bin B$ are adjacent iff there exists a vertex $xin X$ adjacent in $G$ to both $a$ and $b$. We claim that $G'$ has a perfect matching. To show this we have to check that $G'$ satisfies the conditions of Hall’s Marriage Theorem. Let $W$ be a subset of $A$. Then in $G$ there are $7|W|$ edges incident to vertices of $W$. Each of these edges has a unique adjacent edge of a form $(x,b)$ with some $xin X$ and $bin B$. Then clearly $bin N_{G’}(A)$. Since any vertex $bin B$ can participate in at most $7$ pairs $(x,b)$ for some $xin X$,
$|N_{G’}(A)|ge 7|W|/7=|W|$. Let $M$ be a perfect matching for $G’$. Then $M$ consists of $|A|=|B|=9$ edges and for each edge $(a,x)in M$ there exists a vertex $v(e)in X$ such that $(a,x)$ and $(x,b)$ are edges of the graph $G$. It remains to put $Y={ v(e):ein M}$.
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
answered Feb 4 at 20:35
Alex RavskyAlex Ravsky
42.7k32383
42.7k32383
add a comment |
add a comment |
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