We give simpler proofs for both of their results. Examples and special cases Another upper bound Theorem (Brooks, 1941) If G is connected, and is not the complete graph nor an odd cycle, (G)(G). For every two positive integers a and b such that. In fact, much more can be said: Let n be a positive integer. And, there is no possible improvement of any of these bounds. These bounds are easy to check, but they are not the best possible. This is the Nordhaus-Gaddum Theorem: If G is a graph of order n, then. Furthermore, the proofs and are long and tedious. We also looked at some bounds on the chromatic number, and we keep exploring bounds on the chromatic number today. In other words, for any odd g, the question of attainability of μ( g) is answered for all g by our results. In 6, Pan and Zhu have given a function ( g) that gives an upper bound for the circular-chromatic number for every K4 -minor-free graph Gg of odd girth at least g, g 3. We prove that for every odd integer g = 2 k + 1, there exists a graph G g ∈ G/ K 4 of odd girth g such that χ c( G g) = μ( g) if and only if k is not divisible by 3. In, they have shown that their upper bound in can not be improved by constructing a sequence of graphs approaching μ( g) asymptotically. In, Pan and Zhu have given a function μ( g) that gives an upper bound for the circular-chromatic number for every K 4-minor-free graph G g of odd girth at least g, g ≥ 3. We say that the circular chromatic number of G, denoted χ c( G), is equal to the smallest k/ d where a k/ d -coloring exists. ![]() , k - 1, such that d ≤ | c( x) - c( y) | ≤ k - d, whenever xy is an edge of G. One can also employ fancy Lovasz theta-function. There are two obvious: chi (G) geq omega (G) and chi (G) geq n / alpha (G). Im curious what are the known lower bounds for chromatic number. For k ≥ d ≥ 1, a k/ d -coloring of G is a coloring c of vertices of G with colors 0, 1, 2. I am trying to find a good lower bound for chromatic number of one family of graphs.
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