Mixing
Maximal chromatic polynomial of a graph with fixed chromatic number
1
$begingroup$
Consider all the graphs with $n$ vertices and whose chromatic number is $k$, note that the graph does not have to be connected. What are the graphs with the maximal chromatic polynomial among those graphs?
I think that the answer should be the graphs consisting of a $k$-clique plus $n-k$ isolated vertices, is it true? How to prove this?
graph-theory coloring
edited Jan 20 at 3:21
Zach Langley
9731019
asked Jan 20 at 2:55
dendimon22dendimon22
61
$endgroup$
add a comment |
1
$begingroup$
Consider all the graphs with $n$ vertices and whose chromatic number is $k$, note that the graph does not have to be connected. What are the graphs with the maximal chromatic polynomial among those graphs?
I think that the answer should be the graphs consisting of a $k$-clique plus $n-k$ isolated vertices, is it true? How to prove this?
graph-theory coloring
edited Jan 20 at 3:21
Zach Langley
9731019
asked Jan 20 at 2:55
dendimon22dendimon22
61
$endgroup$
$begingroup$
How do you define a maximal polynomial?
$endgroup$
– Leen Droogendijk
Jan 20 at 8:01
$begingroup$
maximal in terms of the grahp with maximum number of colorings
$endgroup$
– dendimon22
Jan 21 at 23:39
$begingroup$
This problem is still unsolved and was stated independently by Wilf and Linial. please cheek this paper : Loh, P. S., Pikhurko, O., & Sudakov, B. (2010). Maximizing the number of q‐colorings. Proceedings of the London Mathematical Society, 101(3), 655-696.
$endgroup$
– W.R.P.S
Jan 22 at 10:42
$begingroup$
@W.R.P.S The unsolved problem is over all $n$-vertex graphs with $m$ edges, without fixing the chromatic number - the version of the problem in this question seems easier.
$endgroup$
– Misha Lavrov
Jan 22 at 19:13
$begingroup$
I see now that this is a pretty hard question and probably an open one, but what if we add an assumption that the graph have an universal vertex? It sounds much easier now, can someone prove this proposition?
$endgroup$
– dendimon22
Jan 23 at 3:07
add a comment |
1
1
1
1
$begingroup$
Consider all the graphs with $n$ vertices and whose chromatic number is $k$, note that the graph does not have to be connected. What are the graphs with the maximal chromatic polynomial among those graphs?
I think that the answer should be the graphs consisting of a $k$-clique plus $n-k$ isolated vertices, is it true? How to prove this?
graph-theory coloring
edited Jan 20 at 3:21
Zach Langley
9731019
asked Jan 20 at 2:55
dendimon22dendimon22
61
$endgroup$
Consider all the graphs with $n$ vertices and whose chromatic number is $k$, note that the graph does not have to be connected. What are the graphs with the maximal chromatic polynomial among those graphs?
I think that the answer should be the graphs consisting of a $k$-clique plus $n-k$ isolated vertices, is it true? How to prove this?
graph-theory coloring
graph-theory coloring
edited Jan 20 at 3:21
Zach Langley
9731019
asked Jan 20 at 2:55
dendimon22dendimon22
61
edited Jan 20 at 3:21
Zach Langley
9731019
asked Jan 20 at 2:55
dendimon22dendimon22
61
edited Jan 20 at 3:21
Zach Langley
9731019
edited Jan 20 at 3:21
Zach Langley
9731019
edited Jan 20 at 3:21
Zach Langley
9731019
9731019
asked Jan 20 at 2:55
dendimon22dendimon22
61
asked Jan 20 at 2:55
dendimon22dendimon22
61
asked Jan 20 at 2:55
dendimon22dendimon22
61
61
$begingroup$
How do you define a maximal polynomial?
$endgroup$
– Leen Droogendijk
Jan 20 at 8:01
$begingroup$
maximal in terms of the grahp with maximum number of colorings
$endgroup$
– dendimon22
Jan 21 at 23:39
$begingroup$
This problem is still unsolved and was stated independently by Wilf and Linial. please cheek this paper : Loh, P. S., Pikhurko, O., & Sudakov, B. (2010). Maximizing the number of q‐colorings. Proceedings of the London Mathematical Society, 101(3), 655-696.
$endgroup$
– W.R.P.S
Jan 22 at 10:42
$begingroup$
@W.R.P.S The unsolved problem is over all $n$-vertex graphs with $m$ edges, without fixing the chromatic number - the version of the problem in this question seems easier.
$endgroup$
– Misha Lavrov
Jan 22 at 19:13
$begingroup$
I see now that this is a pretty hard question and probably an open one, but what if we add an assumption that the graph have an universal vertex? It sounds much easier now, can someone prove this proposition?
$endgroup$
– dendimon22
Jan 23 at 3:07
add a comment |
$begingroup$
How do you define a maximal polynomial?
$endgroup$
– Leen Droogendijk
Jan 20 at 8:01
$begingroup$
maximal in terms of the grahp with maximum number of colorings
$endgroup$
– dendimon22
Jan 21 at 23:39
$begingroup$
This problem is still unsolved and was stated independently by Wilf and Linial. please cheek this paper : Loh, P. S., Pikhurko, O., & Sudakov, B. (2010). Maximizing the number of q‐colorings. Proceedings of the London Mathematical Society, 101(3), 655-696.
$endgroup$
– W.R.P.S
Jan 22 at 10:42
$begingroup$
@W.R.P.S The unsolved problem is over all $n$-vertex graphs with $m$ edges, without fixing the chromatic number - the version of the problem in this question seems easier.
$endgroup$
– Misha Lavrov
Jan 22 at 19:13
$begingroup$
I see now that this is a pretty hard question and probably an open one, but what if we add an assumption that the graph have an universal vertex? It sounds much easier now, can someone prove this proposition?
$endgroup$
– dendimon22
Jan 23 at 3:07
$begingroup$
How do you define a maximal polynomial?
$endgroup$
– Leen Droogendijk
Jan 20 at 8:01
$begingroup$
How do you define a maximal polynomial?
$endgroup$
– Leen Droogendijk
Jan 20 at 8:01
$begingroup$
maximal in terms of the grahp with maximum number of colorings
$endgroup$
– dendimon22
Jan 21 at 23:39
$begingroup$
maximal in terms of the grahp with maximum number of colorings
$endgroup$
– dendimon22
Jan 21 at 23:39
$begingroup$
This problem is still unsolved and was stated independently by Wilf and Linial. please cheek this paper : Loh, P. S., Pikhurko, O., & Sudakov, B. (2010). Maximizing the number of q‐colorings. Proceedings of the London Mathematical Society, 101(3), 655-696.
$endgroup$
– W.R.P.S
Jan 22 at 10:42
$begingroup$
This problem is still unsolved and was stated independently by Wilf and Linial. please cheek this paper : Loh, P. S., Pikhurko, O., & Sudakov, B. (2010). Maximizing the number of q‐colorings. Proceedings of the London Mathematical Society, 101(3), 655-696.
$endgroup$
– W.R.P.S
Jan 22 at 10:42
$begingroup$
@W.R.P.S The unsolved problem is over all $n$-vertex graphs with $m$ edges, without fixing the chromatic number - the version of the problem in this question seems easier.
$endgroup$
– Misha Lavrov
Jan 22 at 19:13
$begingroup$
@W.R.P.S The unsolved problem is over all $n$-vertex graphs with $m$ edges, without fixing the chromatic number - the version of the problem in this question seems easier.
$endgroup$
– Misha Lavrov
Jan 22 at 19:13
$begingroup$
I see now that this is a pretty hard question and probably an open one, but what if we add an assumption that the graph have an universal vertex? It sounds much easier now, can someone prove this proposition?
$endgroup$
– dendimon22
Jan 23 at 3:07
$begingroup$
I see now that this is a pretty hard question and probably an open one, but what if we add an assumption that the graph have an universal vertex? It sounds much easier now, can someone prove this proposition?
$endgroup$
– dendimon22
Jan 23 at 3:07
add a comment |
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$begingroup$
How do you define a maximal polynomial?
$endgroup$
– Leen Droogendijk
Jan 20 at 8:01
$begingroup$
maximal in terms of the grahp with maximum number of colorings
$endgroup$
– dendimon22
Jan 21 at 23:39
$begingroup$
This problem is still unsolved and was stated independently by Wilf and Linial. please cheek this paper : Loh, P. S., Pikhurko, O., & Sudakov, B. (2010). Maximizing the number of q‐colorings. Proceedings of the London Mathematical Society, 101(3), 655-696.
$endgroup$
– W.R.P.S
Jan 22 at 10:42
$begingroup$
@W.R.P.S The unsolved problem is over all $n$-vertex graphs with $m$ edges, without fixing the chromatic number - the version of the problem in this question seems easier.
$endgroup$
– Misha Lavrov
Jan 22 at 19:13
$begingroup$
I see now that this is a pretty hard question and probably an open one, but what if we add an assumption that the graph have an universal vertex? It sounds much easier now, can someone prove this proposition?
$endgroup$
– dendimon22
Jan 23 at 3:07