The 4-Acyclic Edge Coloring of Graphs with Large Girths

Backbone coloring for graphs with large girths

For a graph G and a subgraph H (called backbone graph) of G, a backbone k-coloring of G with respect to H is a proper vertex coloring of G using colors from the set {1, 2, . . . , k}, with an additional condition that colors for any two adjacent vertices in H must differ by at least two. The backbone chromatic number of G over H, denoted by BBC(G,H), is the smallest k of a backbone k-coloring a...

Acyclic edge coloring of graphs

An acyclic edge coloring of a graph G is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The acyclic chromatic index χa(G) of a graph G is the least number of colors needed in any acyclic edge coloring of G. Fiamčík (1978) conjectured that χa(G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G...

Acyclic edge coloring of 2-degenerate graphs

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). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contain...

Acyclic Edge coloring of Planar Graphs

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...

Acyclic List Edge Coloring of Graphs