Group path covering and distance two labeling of graphs

  • Authors:

  • Feng Wang;Wensong Lin

  • Affiliations:

  • Department of Mathematics, Southeast University, Nanjing, 210096, PR China and Department of Business Administration, Shanghai Lixin University of Commerce, Shanghai, 201620, PR China;Department of Mathematics, Southeast University, Nanjing, 210096, PR China

  • Venue:
  • Information Processing Letters
  • Year:
  • 2011

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Abstract

For a positive integer d, an L(d,1)-labeling f of a graph G is an assignment of integers to the vertices of G such that |f(u)-f(v)|=d if uv@?E(G), and |f(u)-f(v)|=1 if u and u are at distance two. The span of an L(d,1)-labeling f of a graph is the absolute difference between the maximum and minimum integers used by f. The L(d,1)-labeling number of G, denoted by @l"d","1(G), is the minimum span over all L(d,1)-labelings of G. An L^'(d,1)-labeling of a graph G is an L(d,1)-labeling of G which assigns different labels to different vertices. Denote by @l"d","1^'(G) the L^'(d,1)-labeling number of G. Georges et al. [Discrete Math. 135 (1994) 103-111] established relationship between the L(2,1)-labeling number of a graph G and the path covering number of G^c, the complement of G. In this paper we first generalize the concept of the path covering of a graph to the t-group path covering. Then we establish the relationship between the L^'(d,1)-labeling number of a graph G and the (d-1)-group path covering number of G^c. Using this result, we prove that @l"2","1^'(G) and @l"3","1^'(G) for bipartite graphs G can be computed in polynomial time.