Note: On the hole index of L(2,1)-labelings of r-regular graphs

  • Authors:

  • Sarah Spence Adams;Matthew Tesch;Denise Sakai Troxell;Bradford Westgate;Cody Wheeland

  • Affiliations:

  • Franklin W. Olin College of Engineering, Olin Hall, Olin Way, Needham, MA 02492, USA;Franklin W. Olin College of Engineering, Olin Hall, Olin Way, Needham, MA 02492, USA;Mathematics and Sciences Division, Babson College, Babson Park, MA 02457, USA;Franklin W. Olin College of Engineering, Olin Hall, Olin Way, Needham, MA 02492, USA;Franklin W. Olin College of Engineering, Olin Hall, Olin Way, Needham, MA 02492, USA

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

Quantified Score

Hi-index 0.04

Visualization

Abstract

An L(2,1)-labeling of a graph G is an assignment of nonnegative integers to the vertices of G so that adjacent vertices get labels at least distance two apart and vertices at distance two get distinct labels. A hole is an unused integer within the range of integers used by the labeling. The lambda number of a graph G, denoted @l(G), is the minimum span taken over all L(2,1)-labelings of G. The hole index of a graph G, denoted @r(G), is the minimum number of holes taken over all L(2,1)-labelings with span exactly @l(G). Georges and Mauro [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] conjectured that if G is an r-regular graph and @r(G)=1, then @r(G) must divide r. We show that this conjecture does not hold by providing an infinite number of r-regular graphs G such that @r(G) and r are relatively prime integers.