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  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)
A dynamic coloring of a graph G is a proper coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant c such that for every k-regular graph G, χ d (G) ≤ χ(G) + c… (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)
There are several topological results ensuring in any properly colored graph the existence of a colorful complete bipartite subgraph, whose order is bounded from below by some topological invariants of some topological spaces associated to the graph. Meunier [Electron. J. Combin., 2014] presented the first colorful type result for uniform hypergraphs. In… (More)
In an earlier paper, the present authors (2013)  introduced the alternating chromatic number for hypergraphs and used Tucker's Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the alternating chromatic number is a lower bound for the chromatic number. In this paper, we determine the chromatic number of some families of… (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.