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+5 votes
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.
asked in Graph Theory by Veteran (40.7k points)  
edited by | 940 views

3 Answers

+15 votes
Best answer

Four colour theorem is famous result, it says that any planar graphs can be coloured with only 4 colours !

Ref ->


Note for confused people => laugh

Here ANY is used in sense of FOR ALL x , so , here ANY means literally any one of graph can be selected !

What are you saying , is something like There exists, but in that case, they will say, That specific graph directly..

Any man alive is gonna die => This means all men are gonna die ! Not specific to anyone !
Hope this clears thing a bit !
answered by Veteran (40.7k points)  
selected by
actually ..I have a question ..null graph is also considerd as planer ..but we can't color null graph with 4 colors?? I think here any is an ambiguous word ..the answer might be 4 but this null graph thing I dont understand..


Hope that you got your answer :) Check answer again !

ok it thanks
i have a question,

planar graphs not in syllabus right ?
I have one doubt the theorem states that no more than 4 colors are required but the ques is asking for minimum, is it not contradictory? I mean if we take two points and join them it can be colored using 2 colors. I am not including single point as disconnected graphs are not considered for coloring. @Akash Kanase @Arjun
even I thought the same ..Null graph is also planer but we can color that by only 1 color..but I think the question here says sufficient to vertex color "any" planer graph ..with just 1 or 2 colors we can't color K4 ..although it would have been much better if they had not used "any" .Any is used as For all x but we should always avoid using Any as it is ambiguous.
Okay so it means min for worst case scenario.. Got it thanks :)
Question is confusing.. At the most 4 colours are used..interpretation of question could vary from person to person :)
But knowing English grammar is also necessary for GATE.
Yeah English and context free too :P
+4 votes
According to four color theorem, Every planar graph can be colored using 4 colors.
answered by Junior (717 points)  
0 votes
@arjun sir : can you verify

as far i know , 4 is neccessary to colour any graph (planar )

but 5 is sufficent one

but since they have asked minimum colour so it should be 4 right ?

Ref : Narisngh deo book
answered by Boss (8.7k points)  
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