The specific case where each edge is assigned a unique color (that is, each link belongs to only one SRLG) is strictly equivalent to the

Labeled Graph model considered by Chang and Leu (1997); Wan et al.

Segev, The complexity of bottleneck

labeled graph problems, In Proc.

By taking the edge labels of a sequentially

labeled graph with q edges modulo q, we obviously obtain a harmoniously

labeled graph.

A

labeled graph can be represented as a quadruple g = <V, E, [l.sub.v], [l.sub.E]>, where V is a set of vertices, and E [subset or equal to] V x V is a set of edges.

Definition 1 (

Labeled Graph) A

labeled graph G is a 4-tuple (V, E, 1, B), where V is the vertex set, E [??] V x V is the edge set, and I is the label function that maps a vertex in V or an edge in E to a label in the label set 8, i.e.

Definition 1 (

Labeled Graph) A

labeled graph G is a five element tuple G = {V, E, [[summation].sub.V], [[summation].sub.E], L} where V is a set of vertices and E [subset not equal to] V x V is a set of undirected edges.

When they formulated questions, wrote reports and letters, and

labeled graphs, my students gained experience in clarifying and verbalizing their ideas and expressing them effectively in writing; they also got practice in spelling, punctuation, and handwriting.

Throughout this paper rewritings of directed, relational, and

labeled graphs ar discussed.

Let [G.sub.n,k] be the set of

labeled graphs defined from the set M.

The asymptotic number of

labeled graphs with given degree sequences.

Let l [member of] [2, [l.sub.0]] and let S [subset] [C.sub.n,m] denote the set of

labeled graphs in [C.sub.n,m] whose number of blocks of size t is not in the interval (1 [+ or -] [epsilon]) [b.sub.l]n.