The L(2, 1)-labeling of K1, n-free graphs and its applications

Abstract An L ( 2 , 1 ) -labeling of a graph G is a function f from the vertex set V ( G ) into the set of nonnegative integers such that | f ( x ) − f ( y ) | ≥ 2 if d ( x , y ) = 1 and | f ( x ) − f ( y ) | ≥ 1 if d ( x , y ) = 2 , where d ( x , y ) denotes the distance between x and y in G . The L ( 2 , 1 ) -labeling number, λ ( G ) , of G is the minimum k where G has an L ( 2 , 1 ) -labeling f with k being the absolute difference between the largest and smallest image points of f . In this work, we will study the L ( 2 , 1 ) -labeling on K 1 , n -free graphs where n ≥ 3 and apply the result to unit sphere graphs which are of particular interest in the channel assignment problem.

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

[2]  Pranava K. Jha,et al.  On L(2, 1)-labeling of the Cartesian product of a cycle and a path , 2000, Ars Comb..

[3]  François Sigrist.  Sphere packing , 1983 .

[4]  Daniel Král,et al.  A Theorem about the Channel Assignment Problem , 2003, SIAM J. Discret. Math..

[5]  John P. Georges,et al.  On the size of graphs labeled with a condition at distance two , 1996, J. Graph Theory.

[6]  Frank Harary,et al.  Graph Theory , 2016 .

[7]  Zhendong Shao,et al.  The L(2,1)-labeling and operations of graphs , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

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

[9]  G. Chang,et al.  Labeling graphs with a condition at distance two , 2005 .

[10]  Roger K. Yeh.  THE EDGE SPAN OF DISTANCE TWO LABELLINGS OF GRAPHS , 2000 .

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  J. Georges,et al.  On the size of graphs labeled with condition at distance two , 1996 .

[13]  Fred S. Roberts,et al.  T-colorings of graphs: recent results and open problems , 1991, Discret. Math..

[14]  Jan van Leeuwen,et al.  lambda-Coloring of Graphs , 2000, STACS.

[15]  Jing-Ho Yan,et al.  On L(2, 1)-labelings of Cartesian products of paths and cycles , 2004, Discret. Math..

[16]  W. K. Hale.  Frequency assignment: Theory and applications , 1980, Proceedings of the IEEE.

[17]  Daphne Der-Fen Liu,et al.  On Distance Two Labellings of Graphs , 1997, Ars Comb..

[18]  David Zhang,et al.  Improved Bounds on the $L(2,1)$-Number of Direct and Strong Products of Graphs , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[19]  Denise Sakai,et al.  Labeling Chordal Graphs: Distance Two Condition , 1994 .

[20]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[21]  John P. Georges,et al.  Edge Labelings with a Condition at Distance Two , 2004, Ars Comb..

[22]  Daphne Der-Fen Liu,et al.  On L(d, 1)-labelings of graphs , 2000, Discret. Math..

[23]  LI Shuang-cheng,et al.  The L(d■,1 ■)-labeling of graphs , 2003 .