Let us suppose that a graph Gn has no triangle and the numbe

Let us suppose that a graph, Gn, has no triangle and the number of its edges is 4n.

• Prove that cr(Gn) 2n.

• ** Use the probabilistic method to give a better bound, cr(Gn) 2.37n.

Solution

We know that the chromatic number Cr(G) is the least k such that G is k-colourable.

Chromatic number is always Cr (G) >= Number of Vetices / stability number , equation 1

here Number of vertices = 4n , Stability number is nothing but the size of maximum independent set.

For triangle-less planar graph , We know stability number is always >= 2. equation 2

From equation 1 and 2 , we get Cr (G) >= 4n/2

Using Probabilistic approach:

By Hamiltian theorm for n no of edges we know Chromatic number is always >= n!2(n1)

so for 4n edges , we have Cr (G) >= 4n! 2^-4n+1 .>= 2 * 4n!/2^{4n ,}

4n!/2⁴ⁿ = (4n-1) (4n-2) (4n-3)../2⁴ⁿ , this >= 1.18n

There fore Cr (G) >= 2 * 1.18n >= 2.37 n