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Recent progress in mathematics and engineering on optimal graph labellings with distance conditions

Author

Listed:
  • Jerrold R. Griggs

    (University of South Carolina)

  • Xiaohua Teresa Jin

    (University of Vermont)

Abstract

The problem of radio channel assignments with multiple levels of interference can be modelled using graph theory. The theory of integer vertex-labellings of graphs with distance conditions has been investigated for several years now, and the authors recently introduced a new model of real number labellings that is giving deeper insight into the problems. Here we present an overview of the recent outpouring of papers in the engineering literature on such channel assignment problems, with the goal of strengthening connections to applications. Secondly, we present a new contribution to the theory, the formulas for the optimal span of labellings with conditions at distance two for finite complete bipartite graphs.

Suggested Citation

  • Jerrold R. Griggs & Xiaohua Teresa Jin, 2007. "Recent progress in mathematics and engineering on optimal graph labellings with distance conditions," Journal of Combinatorial Optimization, Springer, vol. 14(2), pages 249-257, October.
    • Handle: RePEc:spr:jcomop:v:14:y:2007:i:2:d:10.1007_s10878-007-9055-7
    DOI: 10.1007/s10878-007-9055-7
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    References listed on IDEAS

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    1. Xiaohua Teresa Jin & Roger K. Yeh, 2005. "Graph distance‐dependent labeling related to code assignment in computer networks," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(2), pages 159-164, March.
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    Cited by:

    1. Wensong Lin & Benqiu Dai, 2015. "On $$(s,t)$$ ( s , t ) -relaxed $$L(2,1)$$ L ( 2 , 1 ) -labelings of the triangular lattice," Journal of Combinatorial Optimization, Springer, vol. 29(3), pages 655-669, April.
    2. Qiong Wu & Wai Chee Shiu & Pak Kiu Sun, 2014. "Circular L(j,k)-labeling number of direct product of path and cycle," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 355-368, February.
    3. Qiong Wu & Wai Chee Shiu & Pak Kiu Sun, 2016. "$$L(j,k)$$ L ( j , k ) -labeling number of Cartesian product of path and cycle," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 604-634, February.
    4. Pu Zhang & Wensong Lin, 2014. "Multiple L(j,1)-labeling of the triangular lattice," Journal of Combinatorial Optimization, Springer, vol. 27(4), pages 695-710, May.
    5. Xiaoling L. Zhang & Jianguo G. Qian, 2016. "$$L(p,q)$$ L ( p , q ) -labeling and integer tension of a graph embedded on torus," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 67-77, January.
    6. Wensong Lin, 2016. "On $$(s,t)$$ ( s , t ) -relaxed $$L(2,1)$$ L ( 2 , 1 ) -labeling of graphs," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 405-426, January.
    7. Wensong Lin & Chenli Shen, 2018. "Channel assignment problem and n-fold t-separated $$L(j_1,j_2,\ldots ,j_m)$$ L ( j 1 , j 2 , … , j m ) -labeling of graphs," Journal of Combinatorial Optimization, Springer, vol. 35(4), pages 1147-1167, May.
    8. Wensong Lin & Jianzhuan Wu, 2013. "Distance two edge labelings of lattices," Journal of Combinatorial Optimization, Springer, vol. 25(4), pages 661-679, May.

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