Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract   One of the basic results in graph colouring is Brooks' theorem [4] which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Gallai [6] characterized the subgraphs of  k -colour-critical graphs induced by the set of all vertices of degree  k  − 1. The choosability version of Brooks' theorem was proved, independently, by Vizing [9] and by Erdos et al. [5]. As Thomassen pointed out in his talk at the Graph Theory Conference held at Oberwolfach, July 1994, one can also prove a choosability version of Gallai's result.  All these theorems can be easily derived from a result of Borodin [2, 3] and Erdos et al. [5] which enables a characterization of connected graphs  G  admitting a color scheme  L  such that | L ( x )| ⩾  d  G  ( x ) for all  x   ϵ   V  ( G ) and there is no  L -colouring of  G . In this note,  use a reduction idea in order to give a new short proof of this result and to extend it to hypergraphs.
['Alexandr V. Kostochka', 'Michael Stiebitz', 'B. Wirth']
I am writing a research paper on a idea The colour theorems of Brooks and Gallai extended
[{"from": "human", "value": "I am writing a research paper on a idea The colour theorems of Brooks and Gallai extended"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract   One of the basic results in graph colouring is Brooks' theorem [4] which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Gallai [6] characterized the subgraphs of  k -colour-critical graphs induced by the set of all vertices of degree  k  \u2212 1. The choosability version of Brooks' theorem was proved, independently, by Vizing [9] and by Erdos et al. [5]. As Thomassen pointed out in his talk at the Graph Theory Conference held at Oberwolfach, July 1994, one can also prove a choosability version of Gallai's result.  All these theorems can be easily derived from a result of Borodin [2, 3] and Erdos et al. [5] which enables a characterization of connected graphs  G  admitting a color scheme  L  such that | L ( x )| \u2a7e  d  G  ( x ) for all  x   \u03f5   V  ( G ) and there is no  L -colouring of  G . In this note,  use a reduction idea in order to give a new short proof of this result and to extend it to hypergraphs."}]