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Antibandwidth & Cyclic Antibandwidth of Graphs: Exploring Labeling Problems

Explore the concepts of antibandwidth and cyclic antibandwidth in graphs, aiming to maximize the difference between adjacent vertices. With a focus on terminology clarification and presenting upper and lower bounds for meshes, hypercubes, and toroidal meshes, this study provides insights into optimal numbering strategies. Discover how these labeling problems can lead to improved bounds and valuable results in graph theory.

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Antibandwidth & Cyclic Antibandwidth of Graphs: Exploring Labeling Problems

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  1. André Raspaud, Ondrej Sýkora, Heiko Schröder, Ľubomír Török, Imrich Vrťo Antibandwidth and Cyclic Antibandwidth of Meshes and Hypercubes

  2. Dedicated to memory of Ondrej Sýkora V 12.5. 2005

  3. Antibandwidth problem • Consists of placing the vertices of a graph on a line in consecutive integer points in such a way that the minimum difference of adjacent vertices is maximized. • Just another labeling problem (from graph theory point of view)

  4. Confusing terminology • Originally studied under the term separation number(Leung, Vornberger, On some variants of the bandwidth minimization problem) • Dual bandwidth(Lin, Yuan) • We propose (best and hopefully final) term: antibandwidth

  5. Previous results • NP-complete (Leung, Vornberger) • Polynomially solvable for the complements of • Interval • Arborescent comparability • Treshold graphs. (Donnely, Isaak, Hamiltonian powers in … graphs)

  6. Previous results • Exact results for: • Paths, cycles, special trees, complete and complete bipartite graphs • Also interesting for disconnected graphs. • Exact values for graphs consisting of copies of simple graphs.

  7. Previous results • m x n mesh, , (Miller, Pritikin,On the separation number of graphs) • N-dimensional hypercube (Miller, Pritikin,…)

  8. Our contribution • Upper bound method suitable for bipartite graphs • Improving bounds for hypercubes and meshes:

  9. Our contribution • Toroidal meshes : • Even n: • Odd n:

  10. Meshes: upper bound • Definition: Let be a bipartition. Minimal vertex boundary of a set is a set of all vertices from having neighbour in A. • Proof based on result of Bezrukov and Piotrowski (minimal bipartite vertex boundary of mesh)

  11. Meshes: lower bound • Showed in Miller, Pritikin, On separation number of graphs

  12. Even torus • Optimal numbering for even torus

  13. Odd torus • Optimal numbering of odd torus

  14. Hypercube • Vertices of can be partitioned into sets according to their distance from the vertex 00...0. • Edges are only between and .

  15. Cyclic antibandwidth • The vertices are mapped bijectively into such that the minimal distance, measured in cycle, of adjacent vertices is maximized. • We provide: • General lower bound • Values for meshes, tori and hypercubes

  16. General bounds • Upper bound • Lower bound

  17. Cyclic antibandw. of meshes • m even, n odd, then • Otherwise

  18. Cyclic antibandw. of meshes • Another optimal numbering of mesh comparing the antibandwidth part

  19. Cyclic antibandw. of meshes • Way of proof • To show that previous numbering is also ab-optimal • Computing the length of the longest edge in this labeling • Getting the lower bound value from general formula

  20. Cyclic antibandw. of tori • Even torus • Odd torus

  21. Cyclic antibandw. of hypercube • Similar way of proof as for mesh

  22. Conclusion • Antibandwidth • Improved bounds for meshes and hypercubes • Results for toroidal meshes • Cyclic antibandwidth • General bounds based on antibandwidth • Results for meshes, hypercubes and toroidal meshes

  23. Thank You The End

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