Given a nite directed graph, a coloring of its edges turns the graph into a nite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected nite graph of a uniform outdegree (constant outdegree of any vertex) is k-...

We show how graph neural networks can be used to solve the canonical coloring problem. frame as a multiclass node classification problem and utilize an unsupervised training strategy based on statistical physics Potts model. Generalizations other problems such community detection, data clustering, minimum clique cover are straightforward. provide numerical benchmark results illustrate our appro...

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...