We prove the theorem from the title: the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov and Zaks and presents further support for a conjecture that ∆(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analogue of a result of Robinson and the second auth...

A walk W in edge-colored graph is called properly colored (PC) if every pair of consecutive edges in W is of different color. We study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity of type i is a proper superset of graphs of acyclicity of type i+1, i = 1, 2, 3, 4. The first three types are equivalent to the absence of PC cycles, PC trails, and PC walks. Wh...

We consider constrained proper edge colorings of the following type: Given a positive integer j and a family F of connected graphs on 3 or more vertices, we require that the subgraph formed by the union of any j color classes has no copy of any member of F . This generalizes some well-known types of colorings such as acyclic edge colorings, distance-2 edge colorings, low treewidth edge coloring...

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F), is defined as the maximum a(G) over all the graphs G ∈ F . In this pape...

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of a graph G = (V,E). For a family F of graphs, the acyclic chromatic number of F , denoted by a(F ), is defined as the maximum a(G) over all the graphs G ∈ F . In this p...

An acyclic coloring of a graph is a proper coloring such that any two color classes induce a forest. A star coloring of a graph is an acyclic coloring with the further restriction that the forest induced by any two color classes is a disjoint collection of stars. We consider the behavior of these problems when restricted to certain classes of graphs. In particular, we give characterizations of ...

An acyclic coloring of a graph is a proper vertex coloring without bichromatic cycles. We show that the acyclic colorings of any weakly chordal graph G correspond to the proper colorings of triangulations of G. As a consequence, we obtain polynomial-time algorithms for the acyclic coloring problem and the perfect phylogeny problem on the class of weakly chordal graphs. Our results also imply li...

In this article we survey the emerging field of descriptive graph combinatorics. This area has developed in the last two decades or so at the interface of descriptive set theory and graph theory, and it has interesting connections with other areas such as ergodic theory and probability theory. Our object of study is the theory of definable graphs, usually Borel or analytic graphs on Polish spac...