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On the L(2, 1)-labeling conjecture for brick product graphs

Author

Listed:
  • Zehui Shao

    (Chengdu University)

  • Xiaosong Zhang

    (University of Electronic Science and Technology of China)

  • Huiqin Jiang

    (Chengdu University)

  • Bo Wang

    (Chengdu University)

  • Juanjuan He

    (Wuhan University of Science and Technology
    Hubei Province Key Laboratory of Intelligent Information Processing and Real-time Industrial System)

Abstract

Let $$G=(V, E)$$ G = ( V , E ) be a graph. Denote $$d_G(u, v)$$ d G ( u , v ) the distance between two vertices u and v in G. An L(2, 1)-labeling of G is a function $$f: V \rightarrow \{0,1,\ldots \}$$ f : V → { 0 , 1 , … } such that for any two vertices u and v, $$|f(u)-f(v)| \ge 2$$ | f ( u ) - f ( v ) | ≥ 2 if $$d_G(u, v) = 1$$ d G ( u , v ) = 1 and $$|f(u)-f(v)| \ge 1$$ | f ( u ) - f ( v ) | ≥ 1 if $$d_G(u, v) = 2$$ d G ( u , v ) = 2 . The span of f is the difference between the largest and the smallest number in f(V). The $$\lambda $$ λ -number $$\lambda (G)$$ λ ( G ) of G is the minimum span over all L(2, 1)-labelings of G. In this paper, we conclude that the $$\lambda $$ λ -number of each brick product graph is 5 or 6, which confirms Conjecture 6.1 stated in Li et al. (J Comb Optim 25:716–736, 2013).

Suggested Citation

  • Zehui Shao & Xiaosong Zhang & Huiqin Jiang & Bo Wang & Juanjuan He, 2017. "On the L(2, 1)-labeling conjecture for brick product graphs," Journal of Combinatorial Optimization, Springer, vol. 34(3), pages 706-724, October.
    • Handle: RePEc:spr:jcomop:v:34:y:2017:i:3:d:10.1007_s10878-016-0101-1
    DOI: 10.1007/s10878-016-0101-1
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