graphruled

Graphruled is a concept in graph theory describing graphs that admit a labeling of vertices with integers such that every edge joins vertices whose labels differ by exactly one. Formally, a graph G=(V,E) is graphruled if there exists a function f: V -> Z with |f(u) - f(v)| = 1 for every edge {u,v} in E. In finite graphs this is often represented with labels in a finite range {1,...,k} for some k.

Equivalence to bipartite graphs: A graph is graphruled if and only if it is bipartite. If G is graphruled, then parity of f(v) provides a 2-coloring, since adjacent vertices have labels of opposite parity; hence G is bipartite. Conversely, if G is bipartite, one can fix a root in each connected component and label vertices by their distance from the root. For every edge, endpoints differ by exactly one in distance, so the labeling satisfies the graphruled condition.

Consequences and structure: The labeling induces a layered (or leveled) representation of G, with layers corresponding to integer values. The number k of layers can be taken as the maximum distance from the chosen roots plus one; the class includes all trees and even cycles, as well as grid graphs, and more generally all bipartite graphs. Graphruled graphs exclude odd cycles and any non-bipartite structure.

Applications and usage: The concept is used in graph drawing to produce layered layouts and in scheduling and network modeling where a natural flow or precedence aligns with layered levels. It also connects to graph coloring, since a graphruled labeling yields a 2-coloring by parity.

See also: bipartite graph, layered graph, graph coloring.