Parallel concepts in graph theory

Abstract

One edge of a graph G is taken as a basic unit, regarded as the set of its two nodes. Two edges are called parallel (or independent) if they are disjoint. Then a 1-factor (or perfect matching) of G is a spanning set of parallel edges. A 1-factorization of G is a partition of its edge set E( G) into 1-factors, each of which is considered as a color class in a proper edge coloring of G. Then two edge colorings of G are called orthogonal if no two edges with the same first coloring have the same second coloring. Motivated by parallel processing computers, we propose the hierarchy of parallel structures in a graph consisting of (1) an edge, (2) a set of parallel edges, (3) a collection of parallel sets of edges, (4) a class of such orthogonal collections. We also consider similar hierarchies for triangles, nodes, paths, and stars, as well as analogous concepts in other branches of discrete mathematics.