y Abstract We show that if a graph has acyclic chromatic number k, then its game chromatic number is at most k(k + 1). By applying the known upper bounds for the acyclic chromatic numbers of various classes of graphs, we obtain upper bounds for the game chromatic number of these classes of graphs. In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the g...

An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index χa(G) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that χa(G) ≤ ∆(G) + 2 for any simple graph G with maximum degree ∆(G). A graph is 1-planar if it can be drawn on the plane such that every edg...

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χ′a(G), is the least number of colors k such that G has an acyclic edge k-coloring. Basavaraju et al. [Acyclic edgecoloring of planar graphs, SIAM J. Discrete Math. 25 (2) (2011), 463–478] showed that χ′a(G) ≤ ∆(G) + 12 for planar graphs G with maximum degree...

A coloring of a graph G is an acyclic coloring if the union of any two color classes induces a forest. It is proved that the acyclic chromatic number of direct product of two trees T1 and T2 equals min{∆(T1)+1, ∆(T2)+1}. We also prove that the acyclic chromatic number of direct product of two complete graphs Km and Kn is mn − m − 2, where m ≥ n ≥ 4. Several bounds for the acyclic chromatic numb...

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of products of paths and cycles are considered. We determine the acyclic chromatic numbers of three such products: grid graphs...

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamčik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a(G) ≤ ∆ + 2 for any simple graph G with maximum degree ∆. Basavaraju and Chandran (2009) show...

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χa(G) is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and ∆(G) is large enough then χa(G) = ∆(G). We settle this conjecture for planar gr...

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a(G). It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a(G) ≤ ∆ + 2, where ∆ = ∆(G) denotes the maximum degree of the gra...

A proper edge coloring of a graph G is called acyclic if there is no bichromatic cycle in G. The acyclic chromatic index of G, denoted by χa(G), is the least number of colors k such that G has an acyclic edge k-coloring. The maximum average degree of a graph G, denoted by mad(G), is the maximum of the average degree of all subgraphs of G. In this paper, it is proved that if mad(G) < 4, then χa(...

A proper edge-colouring with the property that every cycle contains edges of at least three distinct colours is called an acyclic edge-colouring. The acyclic chromatic index of a graph G, denoted χa(G), is the minimum k such that G admits an acyclic edge-colouring with k colours. We conjecture that if G is planar and ∆(G) is large enough then χa(G) = ∆(G). We settle this conjecture for planar g...