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Minimum number of edges in a graph of order n with chromatic number k

  1. Oct 12, 2011 #1
    What is the minimum number of edges in a [itex]k[/itex]-chromatic connected graph of order [itex]n[/itex]?

    I have read somewhere that this number is equal to [itex]\left(\begin{array}{c} k \\2 \end{array} \right) +n -k [/itex]

    I was thinking about using [itex]K_k[/itex], the complete graph with [itex]k[/itex] vertices. We know that the number of edges in [itex]K_k[/itex] is equal to [itex]\left(\begin{array}{c} k \\2 \end{array} \right)[/itex].

    Also, I wanted to use the fact that if [itex]\chi (M) = k[/itex] i.e. the chromatic number of a graph [itex]M[/itex] is equal to [itex]k[/itex], and if a vertex [itex]v[/itex] has degree strictly less than [itex]k[/itex], then the chromatic number of ( [itex]M[/itex] together with [itex]v[/itex] and its edges) is also equal to [itex]k.[/itex]

    I am not sure how start, I need help organizing my ideas.
     
  2. jcsd
  3. Oct 22, 2011 #2
    Did you solve it?

    One approach could be to prove Hadwiger's conjecture, then it would be an easy corollary.

    HTH
     
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