A Theory for Coloring Walks in a Digraph
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Abstract:
Consider edge colorings of directed graphs where edges of the form v 1 v 2 and v 2 v 3 must have different colors. Here, v 1 � � v 2, v 2 � � v 3 but v 1 = v 3 is possible. It is known that this coloring induces a vertex coloring by sets of edge colors, in which edge v 1 v 2 in the graph implies that the set color of v 1 contains an element not in the set color of v 2; conversely, each such set coloring of vertices induces one or more edge colorings. We show that these relationships generalize to colorings of of k(vertex)-walks in which two k-walks have different colors if one is the prefix and the other is the suffix of a common (k+1)-walk. For full generality the colors belong to a partially ordered set P; and the prefix color c 1 and the suffix color c 2 must satisfy c 1 �� | c 2. The set color construction generalizes to generating the lower order ideal in P from a set of k-walk colors; these order ideals (antichains in P, equivalently) are partially ordered by containment. We conclude that a P coloring of k-walks exists if and only if there is a vertex coloring by A k-1 (P), where A is the operator that maps a poset to its poset of lower order ideals, due to Birkhoff. In the case when the graph G is symmetric, this condition means that the largest antichain size (Dilworth
Citations
| 200 | Data parallel algorithms – Hillis, Steele - 1986 |
| 149 | editor. A Synthesis of Parallel Algorithms – Reif - 1993 |
| 113 | Parallel tree contraction and its applications – Miller, Reif - 1985 |
| 84 | Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking – Cole, Vishkin - 1997 |
| 83 | Combinatorial Theory – Aigner - 1979 |
| 47 | Parallel symmetrybreaking in sparse graphs – Goldberg, Plotkin, et al. - 1987 |
| 30 | The structure of Sperner k-families – Greene, Kleitman - 1976 |
| 25 | Lattice Theory, 3rd edition – BIRKHOFF - 1967 |
| 21 | How robust is the n-cube – Becker, Simon - 1988 |
| 7 | On unavoidaable digraphs in orientation of graphs – Bloom, Burr - 1987 |
| 5 | On directed paths and circuits, in: Theory of Graphs – Gallai - 1968 |
| 1 | Monochromatic paths and circuits in edge-colored graphs – Busolini - 1971 |
| 1 | atal, Monochromatic paths in edge-colored graphs – Chv - 1972 |
