This is inspired by the similar question on modern set theory. I've read through the open problems in graph theory on Wikipedia's list of unsolved problems in mathematics, but what I'm looking for is more an accessible description of the main areas of graph theoretical research. I hope this is okay to ask.
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Someone should ask this about Number Theory. Preferably someone with some sort of relevant information on current hot areas. – Sandeep Silwal Aug 16 '14 at 00:09
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I am unsure whether you are actually interested in subareas of graph theory or in open problems. If you are interested in subareas, Wikipedia lists 5 of them in its article about Graph Theory. Are their descriptions not accessible enough? – F.U.A.S. Oct 07 '22 at 19:13
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The biggest one and the most simple one is the absence of a necessary and sufficient condition for a graph to be Hamiltonian. There are a few sufficient conditions (as far as I know) but all of them are pretty weak - or require strong conditions. The simplicity of Eulerian paths in graphs and the complexity of Hamiltonian paths is fascinating. And if I'm right this is one of the biggest problems in Graph Theory right now.
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