An extremal problem on non-full colorable graphs

  • Authors:

  • Changhong Lu;Mingqing Zhai

  • Affiliations:

  • Department of Mathematics, East China Normal University, Shanghai 200062, China and Institute of Theoretical Computing, ECNU, Shanghai 200062, China;Department of Mathematics, East China Normal University, Shanghai 200062, China and Department of Mathematics, Chuzhou University, Anhui 239012, China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2007

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Abstract

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|≥2 when dG(u,v)=1 and |f(u)-f(v)|≥1 when dG(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].