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Peter Heinig
Peter Heinig
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The OP is asking for something which has already been proved to exist in

Saieed Akbari, Vahid Liaghat, Afshin Nikzad: Colorful Paths in Vertex Coloring of Graphs, The electronic journal of combinatorics 18 (2011), #P17

(see Theorem 13 on page 25 of op. cit., which is exactly what you arethe opening post is asking for)

Little more needs to be said on the question in the OP as it currently stands, I think. If one takesexcept for a (reasearch question)$\Rightarrow$(answer unknown),necessary exclusion of then this is not a research question$G\cong C^7$.)

The OP is asking for something which has already been proved to exist in

Saieed Akbari, Vahid Liaghat, Afshin Nikzad: Colorful Paths in Vertex Coloring of Graphs, The electronic journal of combinatorics 18 (2011), #P17

(see Theorem 1 on page 2, which is exactly what you are asking for)

Little more needs to be said on the question in the OP as it currently stands, I think. If one takes (reasearch question)$\Rightarrow$(answer unknown), then this is not a research question.

The OP is asking for something which has already been proved to exist in

Saieed Akbari, Vahid Liaghat, Afshin Nikzad: Colorful Paths in Vertex Coloring of Graphs, The electronic journal of combinatorics 18 (2011), #P17

(see Theorem 3 on page 5 of op. cit., which is exactly what the opening post is asking for, except for a necessary exclusion of $G\cong C^7$.)

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Peter Heinig
Peter Heinig
  • 6.1k
  • 1
  • 27
  • 47

The OP is asking for something which has already been proved to exist in

Saieed Akbari, Vahid Liaghat, Afshin Nikzad: Colorful Paths in Vertex Coloring of Graphs, The electronic journal of combinatorics 18 (2011), #P17

(see Theorem 1 on page 2, which is exactly what you are asking for)

Little more needs to be said on the question in the OP as it currently stands, I think. If one takes (reasearch question)$\Rightarrow$(answer unknown), then this is not a research question.