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I'm not sure whether this is the right place for this question, but what are the most major unsolved problems in graph theory? (Not just a list, but something like a top 10 list or something like that)

My impression seems to be: - Hadwiger Conjecture - Reconstruction Conjecture - Graceful Tree Conjecture - Tutte's Flow Conjectures

are amongst the biggest. I.e. would gather most superstardom if proved.

Martin Sleziak
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John Smith
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    The "Hamiltonian path" problem in graph theory is NP complete: https://en.wikipedia.org/wiki/Hamiltonian_path – Michael Jul 21 '16 at 02:48
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    It is also unknown whether or not there exists a graph that has a number of nodes equal to an odd perfect number. Haha. – Michael Jul 21 '16 at 02:51
  • @Michael Also whether for every even $n \ge 4$, there exists a bipartite graph of order $n$ with each partite set's size a prime number :þ – M. Vinay Jul 21 '16 at 04:00
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    Related: What are the big issues in modern graph theory? (However, no upvoted answers were posted there.) – Martin Sleziak Jul 21 '16 at 04:26
  • @MartinSleziak : Well I just upvoted your link as it was the same as my first comment. – Michael Jul 21 '16 at 13:23
  • @M.Vinay : Thanks for getting my joke. However, I don't get yours! What does "order $n$" mean and what does "each partite set" mean? It sounds like you are describing a bipartite graph with $n$ nodes on the left and $n$ nodes on the right, where $n$ is an even prime (in which case it is certainly known that $n$ must be 2). – Michael Jul 21 '16 at 13:27
  • @Michael The number of nodes of a graph is usually called the order of the graph. For each even integer $n \ge 4$, does there exist a bipartite graph with a total of $n$ vertices, and whose bipartiton consists of $p$ vertices on the left (as you put it) and $q$ vertices on the right, $p$ and $q$ being prime numbers? – M. Vinay Jul 21 '16 at 13:32
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    @M.Vinay : Ah....Goldbach! – Michael Jul 21 '16 at 13:36
  • One measure of important open problems in graph theory can be gathered by checking the list at http://www.openproblemgarden.org/category/graph_theory and filtering by the importance ( = number of stars) of the listed problems. – Jernej Jul 22 '16 at 13:11
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    One very big problem would be determining the rate of growth of Ramsey numbers. $R(k)$ is defined as the smallest $n$ such that every graph on $n$ vertices has either a clique or an independent set of size $k$. Our lower bounds are essentially $\sqrt{2}^k \le R(k) \le 4^k$; there have been polynomial improvements on the bounds, but nobody has been able to improve either exponential base in about 70 years (which is a long time in graph theory). – Shagnik Jul 23 '16 at 18:08
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    Another area which has attracted a fair amount of attention is understanding why graphs have large chromatic number. Obviously having a large clique forces a large chromatic number, but there are triangle-free graphs with arbitrarily large chromatic numbers. In fact, there are graphs with large girth (i.e. no short cycles) and large chromatic numbers. I'm not sure what single conjecture/problem would best represent this line of research, but I believe Chudnovsky-Scott-Seymour have solve some conjectures of Gyárfás in this direction. – Shagnik Jul 23 '16 at 18:11

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For the sake of providing an answer to this question, Wikipedia has a nice list of (some of) the big unsolved problems in graph theory.

Mike Pierce
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