A note on collections of graphs with non-surjective lambda labelings

The λ-number of a graph G, denoted λ(G), is the smallest integer k such that there exists a function from V(G) into {0, 1, 2,..., k} under which adjacent vertices receive integers which differ by at least 2 and vertices at distance two receive integers which differ by at least 1. We establish the infinitude of the collection of connected graphs G with fixed maximum degree Δ ≥ 4 and fixed λ-number Δ + t, 1 ≤ t ≤ Δ - 1 such that no λ-labeling of G into {0, 1, 2,...., λ(G)} is surjective. Also, from among graphs with no surjective λ-labelings, we construct connected graphs with maximum degree 3, λ-number 5 and arbitrarily large order.

[1]  Peter C. Fishburn,et al.  No-hole L(2, 1)-colorings , 2003, Discret. Appl. Math..

[2]  Denise Sakai Troxell.  Labeling Chordal Graphs: Distance Two Condition , 1994, SIAM Journal on Discrete Mathematics.

[3]  Peter C. Fishburn,et al.  Full Color Theorems for L(2, 1)-Colorings , 2006, SIAM J. Discret. Math..

[4]  John P. Georges,et al.  Relating path coverings to vertex labellings with a condition at distance two , 1994, Discret. Math..

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

[6]  SakaiDenise.  Labeling Chordal Graphs , 1994 .

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

[8]  Frank Harary,et al.  Radio labelings of graphs , 2001 .

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

[10]  John P. Georges,et al.  Labeling Products of Complete Graphs with a Condition at Distance Two , 2001, SIAM J. Discret. Math..

[11]  Andrzej Pelc,et al.  Labeling trees with a condition at distance two , 2003, Electron. Notes Discret. Math..

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

[13]  Jan van den Heuvel,et al.  Graph labeling and radio channel assignment , 1998, J. Graph Theory.

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

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

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

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

[18]  John P. Georges,et al.  On the Structure of Graphs with Non-Surjective L(2, 1)-Labelings , 2005, SIAM J. Discret. Math..

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