Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A vertex colouring c of a graph G is called nonrepetitive if for every integer r ≥ 1 and every path P = (v1,v2,…,v2r) in G, the first half of P is coloured differently from the second half of P, that is, c(vj)≠c(vr+j) for some j = 1,2,…,r. This notion was inspired by a striking result of Thue asserting that the path Pn on n vertices has a nonrepetitive three-colouring, no matter how large n is. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The Thue choice number of G, denoted by πch(G), is the least integer k such that for every k-list assignment L there is a nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. Using the Lefthanded Local Lemma  prove that πch(Pn) ≤ 4 for every n. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 © 2011 Wiley Periodicals, Inc.
['Jarosław Grytczuk', 'Jakub Przybyło', 'Xuding Zhu']
I am writing a research paper on a idea Nonrepetitive list colourings of paths
[{"from": "human", "value": "I am writing a research paper on a idea Nonrepetitive list colourings of paths"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: A vertex colouring c of a graph G is called nonrepetitive if for every integer r \u2265 1 and every path P = (v1,v2,\u2026,v2r) in G, the first half of P is coloured differently from the second half of P, that is, c(vj)\u2260c(vr+j) for some j = 1,2,\u2026,r. This notion was inspired by a striking result of Thue asserting that the path Pn on n vertices has a nonrepetitive three-colouring, no matter how large n is. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The Thue choice number of G, denoted by \u03c0ch(G), is the least integer k such that for every k-list assignment L there is a nonrepetitive colouring c of G satisfying c(v) \u2208 L(v) for every vertex v of G. Using the Lefthanded Local Lemma  prove that \u03c0ch(Pn) \u2264 4 for every n. \u00a9 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2011 \u00a9 2011 Wiley Periodicals, Inc."}]