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Let G be a graph. A proper vertex coloring of G is said to be a dynamic coloring if for every v ∈ V (G) of degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number of G and is denoted by χ 2 (G). It was conjectured that if G is an r-regular… (More)

In this paper, we investigate circular chromatic number of Mycielski construction of graphs. It was shown in [20] that t th Mycielskian of the Kneser graph KG(m, n) has the same circular chromatic number and chromatic number provided that m + t is an even integer. We prove that if m is large enough, then χ(M t (KG(m, n))) = χ c (M t (KG(m, n))) where M t is… (More)

It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if G is a connected graph distinct from C 7 , then there is a χ(G)-coloring of G in which every vertex v ∈ V (G) is an initial vertex of a path P with χ(G) vertices whose colors are different. In [S. Akbari, V. Liaghat, and A.… (More)

The chromatic sum Σ(G) of a graph G is the smallest sum of colors among of proper coloring with the natural number. In this paper, we introduce a necessary condition for the existence of graph homomorphisms. Also, we present Σ(G) < χ f (G)|G| for every graph G.

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