![]() ![]() We give simpler proofs for both of their 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. ![]() POWELL, An Upper Bound for the Chromatic Number of a Graph and Its Application to Timetabling Problems, Comput. Upper bound for chromatic number of partitioned graph. Furthermore, the proofs and are long and tedious. BOUNDS FOR THE CHROMATIC NUMBER OF A GRAPH 97 PROOV: G has chromatic number x(G) and therefore the set V(G). In other words, for any odd g, the question of attainability of μ( g) is answered for all g by our results. 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. We apply these bounds to cycle graph C n and hypercube Q n and show that it is sharp when k is close to the diameter of these graphs. k-chromatic number for general graphs and in consequence a coloring scheme depending on a partition of the vertex set. In, they have shown that their upper bound in can not be improved by constructing a sequence of graphs approaching μ( g) asymptotically. In this article, we concentrate on finding upper bounds of radio. ![]() 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. The harmonious chromatic number of a graph G, denoted by h ( G ), is the least number of colors which can be assigned to the vertices of G such that adjacent vertices are colored differently and any two distinct edges have different color pairs. 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. For k ≥ d ≥ 1, a k/ d -coloring of G is a coloring c of vertices of G with colors 0, 1, 2. ![]()
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