(2, 1)-total Labeling of Trees with Large Maximum Degree

A k -(2, 1)-total labeling of a graph G is to label the vertices and the edges of G using integers from 0 to k such that all adjacent vertices as well as edges receive different labels, and the difference between the labels of a vertex and its incident edges is at least 2. The ( 2 , 1 ) -total labeling number λ 2 t ( G ) is the smallest integer k such that G has a k -(2, 1)-total labeling. It is known that λ 2 t ( T ) , where T is a tree with maximum degree Δ , equals to either Δ + 1 or Δ + 2 . In this paper, we provide a sufficient condition for a tree T to have λ 2 t ( T ) = Δ + 1 when Δ ? 9 .

[1]  Jin Ho Kwak,et al.  Classification of regular embeddings of hypercubes of odd dimension , 2007, Discret. Math..

[2]  Wei-Fan Wang,et al.  (2, 1)-Total number of trees with maximum degree three , 2009, Inf. Process. Lett..

[3]  John P. Georges,et al.  On the lambda-Number of Qn and Related Graphs , 1995, SIAM J. Discret. Math..

[4]  Wei-Fan Wang,et al.  (2, 1)-Total labelling of outerplanar graphs , 2007, Discret. Appl. Math..

[5]  Ko-Wei Lih,et al.  Labeling Planar Graphs with Conditions on Girth and Distance Two , 2004, SIAM J. Discret. Math..

[6]  Jerrold R. Griggs,et al.  Labelling Graphs with a Condition at Distance 2 , 1992, SIAM J. Discret. Math..

[7]  Wei-Fan Wang.  The L(2, 1)-labelling of trees , 2006, Discret. Appl. Math..

[8]  Arnfried Kemnitz,et al.  [r, s, t]-Colorings of graphs , 2007, Discret. Math..

[9]  Jeong-Hyun Kang,et al.  L(2, 1)-Labeling of Hamiltonian graphs with Maximum Degree 3 , 2008, SIAM J. Discret. Math..

[10]  André Raspaud,et al.  (d,1)-total labeling of graphs with a given maximum average degree , 2006 .

[11]  Gerard J. Chang,et al.  The L(2, 1)-Labeling Problem on Graphs , 1996, SIAM J. Discret. Math..

[12]  Mohammad R. Salavatipour,et al.  A bound on the chromatic number of the square of a planar graph , 2005, J. Comb. Theory, Ser. B.

[13]  Frédéric Havet,et al.  (p, 1)-Total labelling of graphs , 2008, Discret. Math..

[14]  Ko-Wei Lih,et al.  On (d, 1)-total numbers of graphs , 2009, Discret. Math..

[15]  Dong Chen,et al.  (2, 1)-Total labelling of trees with sparse vertices of maximum degree , 2009, Inf. Process. Lett..

[16]  Hirotaka Ono,et al.  The (p, q)-total labeling problem for trees , 2012, Discret. Math..