Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A digraph is  k -traceable if its order is at least  k  and each of its subdigraphs of order  k  is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so  k -traceable oriented graphs may be regarded as generalized tournaments. It is ll-known that all tournaments are traceable.  denote by  t ( k ) the smallest integer bigger than or equal to  k  such that every  k -traceable oriented graph of order at least  t ( k ) is traceable. The Traceability Conjecture states that  t ( k ) ≤  2k-1  for every  k  ≥  2  [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008].  show that for  k  ≥ 2, every  k -traceable oriented graph with independence number 2 and order at least 4 k- 12 is traceable. This is the last open case in giving an upper bound for  t ( k ) that is linear in  k .
['Susan A. van Aardt', 'Jean E. Dunbar', 'Marietjie Frick', 'Nicolas Lichiardopol']
I am writing a research paper on a idea A Linear Bound towards the Traceability Conjecture
[{"from": "human", "value": "I am writing a research paper on a idea A Linear Bound towards the Traceability Conjecture"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A digraph is  k -traceable if its order is at least  k  and each of its subdigraphs of order  k  is traceable. An oriented graph is a digraph without 2-cycles. The 2-traceable oriented graphs are exactly the nontrivial tournaments, so  k -traceable oriented graphs may be regarded as generalized tournaments. It is ll-known that all tournaments are traceable.  denote by  t ( k ) the smallest integer bigger than or equal to  k  such that every  k -traceable oriented graph of order at least  t ( k ) is traceable. The Traceability Conjecture states that  t ( k ) \u2264  2k-1  for every  k  \u2265  2  [van Aardt, Dunbar, Frick, Nielsen and Oellermann, A traceability conjecture for oriented graphs, Electron. J. Combin., 15(1):#R150, 2008].  show that for  k  \u2265 2, every  k -traceable oriented graph with independence number 2 and order at least 4 k- 12 is traceable. This is the last open case in giving an upper bound for  t ( k ) that is linear in  k ."}]