J4 ›› 2011, Vol. 46 ›› Issue (4): 9-16.

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The L(p,q)-labeling of planar graphs with girth g(G)≥6

ZHU Hai-yang1, LVXin-zhong2, SHENG Jing-jun1, HANG Dan3   

  1. 1. Department of Logistics Command, Xuzhou Air Force College, Xuzhou 221000, Jiangsu, China;
    2. College of Mathematics, Physics and Information Engineering Zhejiang Normal University, Jinhua 321004, Zhejiang, China;
    3. Department of Basic Courses, Xuzhou Air Force College, Xuzhou 221000, Jiangsu, China
  • Received:2010-04-06 Published:2011-04-21

Abstract:

Let p and q be two positive integers with p≥q, and let λp,q(G) be the L(p,q)-labeling number of a planar graph G. It is proved that λp,q(G)≤(2q-1)Δ(G)+4p+6q-5 if G is a planar graphs with girth g(G)≥6,  and that λp,q(G)≤(2q-1)Δ(G)+10p-2q-4 if G is a planar graphs with girth g(G)≥6 and Δ(G)≠5, which implies that Wegner’s conjecture holds for a planar graph G with girth g(G)≥6 and Δ(G)≠5.

Key words:  planar graph; girth; L(p,q)-labeling; L(p,q)-labeling number

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