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 b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b-vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In thi...

Graph coloring has been studied for a long time and continues to receive interest within the research community [43]. It has applications in scheduling [46], timetables, and compiler register allocation [45]. The most popular variant of graph coloring, k-coloring, can be thought of as an assignment of k colors to the vertices of a graph such that adjacent vertices are assigned different colors....