What are the triangle free graphs on $lfloorfrac{n^2}{4}rfloor$ edges .
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I tried for $n=4$, it is a cycle of length $4$ which is $K_{2,2}$. $n=5$, it seems to be $K_{2,3}$. So my guess is it is a complete bipartite graph?
graph-theory
asked Dec 10 '18 at 15:51
user42493user42493
1877
$endgroup$
add a comment |
$begingroup$
I tried for $n=4$, it is a cycle of length $4$ which is $K_{2,2}$. $n=5$, it seems to be $K_{2,3}$. So my guess is it is a complete bipartite graph?
graph-theory
asked Dec 10 '18 at 15:51
user42493user42493
1877
$endgroup$
1
$begingroup$
That's a strong conjecture on slim evidence. Did you try $n=6$?
$endgroup$
– Lee Mosher
Dec 10 '18 at 15:59
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$K_{3,3}$ has 9 edges
$endgroup$
– user42493
Dec 10 '18 at 16:35
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A bipartite graph is triangle-free, so it's easy to find complete bipartite graphs that satisfy the requirement. The difficulty would be in showing these are the only examples (if that's true.)
$endgroup$
– saulspatz
Dec 10 '18 at 16:36
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I went through the graph on 6 vertices on this link: graphclasses.org/smallgraphs.html#nodes6 and did not find any . I don't know if the list is complete as I didn't find $k_{3,3}$. I couldn't find a good one for 7 vertices either.
$endgroup$
– user42493
Dec 10 '18 at 17:08
add a comment |
$begingroup$
I tried for $n=4$, it is a cycle of length $4$ which is $K_{2,2}$. $n=5$, it seems to be $K_{2,3}$. So my guess is it is a complete bipartite graph?
graph-theory
asked Dec 10 '18 at 15:51
user42493user42493
1877
$endgroup$
I tried for $n=4$, it is a cycle of length $4$ which is $K_{2,2}$. $n=5$, it seems to be $K_{2,3}$. So my guess is it is a complete bipartite graph?
graph-theory
graph-theory
asked Dec 10 '18 at 15:51
user42493user42493
1877
asked Dec 10 '18 at 15:51
user42493user42493
1877
asked Dec 10 '18 at 15:51
user42493user42493
1877
asked Dec 10 '18 at 15:51
user42493user42493
1877
asked Dec 10 '18 at 15:51
user42493user42493
1877
1877
1
$begingroup$
That's a strong conjecture on slim evidence. Did you try $n=6$?
$endgroup$
– Lee Mosher
Dec 10 '18 at 15:59
$begingroup$
$K_{3,3}$ has 9 edges
$endgroup$
– user42493
Dec 10 '18 at 16:35
$begingroup$
A bipartite graph is triangle-free, so it's easy to find complete bipartite graphs that satisfy the requirement. The difficulty would be in showing these are the only examples (if that's true.)
$endgroup$
– saulspatz
Dec 10 '18 at 16:36
$begingroup$
I went through the graph on 6 vertices on this link: graphclasses.org/smallgraphs.html#nodes6 and did not find any . I don't know if the list is complete as I didn't find $k_{3,3}$. I couldn't find a good one for 7 vertices either.
$endgroup$
– user42493
Dec 10 '18 at 17:08
add a comment |
1
$begingroup$
That's a strong conjecture on slim evidence. Did you try $n=6$?
$endgroup$
– Lee Mosher
Dec 10 '18 at 15:59
$begingroup$
$K_{3,3}$ has 9 edges
$endgroup$
– user42493
Dec 10 '18 at 16:35
$begingroup$
A bipartite graph is triangle-free, so it's easy to find complete bipartite graphs that satisfy the requirement. The difficulty would be in showing these are the only examples (if that's true.)
$endgroup$
– saulspatz
Dec 10 '18 at 16:36
$begingroup$
I went through the graph on 6 vertices on this link: graphclasses.org/smallgraphs.html#nodes6 and did not find any . I don't know if the list is complete as I didn't find $k_{3,3}$. I couldn't find a good one for 7 vertices either.
$endgroup$
– user42493
Dec 10 '18 at 17:08
1
1
$begingroup$
That's a strong conjecture on slim evidence. Did you try $n=6$?
$endgroup$
– Lee Mosher
Dec 10 '18 at 15:59
$begingroup$
That's a strong conjecture on slim evidence. Did you try $n=6$?
$endgroup$
– Lee Mosher
Dec 10 '18 at 15:59
$begingroup$
$K_{3,3}$ has 9 edges
$endgroup$
– user42493
Dec 10 '18 at 16:35
$begingroup$
$K_{3,3}$ has 9 edges
$endgroup$
– user42493
Dec 10 '18 at 16:35
$begingroup$
A bipartite graph is triangle-free, so it's easy to find complete bipartite graphs that satisfy the requirement. The difficulty would be in showing these are the only examples (if that's true.)
$endgroup$
– saulspatz
Dec 10 '18 at 16:36
$begingroup$
A bipartite graph is triangle-free, so it's easy to find complete bipartite graphs that satisfy the requirement. The difficulty would be in showing these are the only examples (if that's true.)
$endgroup$
– saulspatz
Dec 10 '18 at 16:36
$begingroup$
I went through the graph on 6 vertices on this link: graphclasses.org/smallgraphs.html#nodes6 and did not find any . I don't know if the list is complete as I didn't find $k_{3,3}$. I couldn't find a good one for 7 vertices either.
$endgroup$
– user42493
Dec 10 '18 at 17:08
$begingroup$
I went through the graph on 6 vertices on this link: graphclasses.org/smallgraphs.html#nodes6 and did not find any . I don't know if the list is complete as I didn't find $k_{3,3}$. I couldn't find a good one for 7 vertices either.
$endgroup$
– user42493
Dec 10 '18 at 17:08
add a comment |
1 Answer
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$begingroup$
This is a partial case of Turán’s theorem for $r=2$. The unique (up to a isomorphism) edge-maximal triangle-free graph is a Turán graph, which is a complete bipartite graph on $n$ vertices such that the sizes of the parts differs by at most one.
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
add a comment |
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$begingroup$
This is a partial case of Turán’s theorem for $r=2$. The unique (up to a isomorphism) edge-maximal triangle-free graph is a Turán graph, which is a complete bipartite graph on $n$ vertices such that the sizes of the parts differs by at most one.
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
add a comment |
$begingroup$
This is a partial case of Turán’s theorem for $r=2$. The unique (up to a isomorphism) edge-maximal triangle-free graph is a Turán graph, which is a complete bipartite graph on $n$ vertices such that the sizes of the parts differs by at most one.
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
add a comment |
$begingroup$
This is a partial case of Turán’s theorem for $r=2$. The unique (up to a isomorphism) edge-maximal triangle-free graph is a Turán graph, which is a complete bipartite graph on $n$ vertices such that the sizes of the parts differs by at most one.
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
$endgroup$
This is a partial case of Turán’s theorem for $r=2$. The unique (up to a isomorphism) edge-maximal triangle-free graph is a Turán graph, which is a complete bipartite graph on $n$ vertices such that the sizes of the parts differs by at most one.
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
answered Dec 10 '18 at 18:33
Alex RavskyAlex Ravsky
41.5k32282
41.5k32282
add a comment |
add a comment |
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1
$begingroup$
That's a strong conjecture on slim evidence. Did you try $n=6$?
$endgroup$
– Lee Mosher
Dec 10 '18 at 15:59
$begingroup$
$K_{3,3}$ has 9 edges
$endgroup$
– user42493
Dec 10 '18 at 16:35
$begingroup$
A bipartite graph is triangle-free, so it's easy to find complete bipartite graphs that satisfy the requirement. The difficulty would be in showing these are the only examples (if that's true.)
$endgroup$
– saulspatz
Dec 10 '18 at 16:36
$begingroup$
I went through the graph on 6 vertices on this link: graphclasses.org/smallgraphs.html#nodes6 and did not find any . I don't know if the list is complete as I didn't find $k_{3,3}$. I couldn't find a good one for 7 vertices either.
$endgroup$
– user42493
Dec 10 '18 at 17:08