Vol. 6, No. 3, 2013

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Rank numbers of graphs that are combinations of paths and cycles

Brianna Blake, Elizabeth Field and Jobby Jacob

Vol. 6 (2013), No. 3, 369–381

A k-ranking of a graph G is a function f : V (G) {1,2,,k} such that if f(u) = f(v), then every u-v path contains a vertex w such that f(w) > f(u). The rank number of G, denoted χr(G), is the minimum k such that a k-ranking exists for G. It is shown that given a graph G and a positive integer t, the question of whether χr(G) t is NP-complete. However, the rank number of numerous families of graphs have been established. We study and establish rank numbers of some more families of graphs that are combinations of paths and cycles.

ranking, $k$-ranking, rank number, paths, cycles
Mathematical Subject Classification 2010
Primary: 05C15, 05C78
Secondary: 05C38
Received: 25 April 2013
Accepted: 29 July 2013
Published: 8 September 2013

Communicated by Joseph A. Gallian
Brianna Blake
Mathematics Department
Augsburg College
Minneapolis, MN 55454
United States
Elizabeth Field
Department of Mathematics
Southern Connecticut State University
New Haven, CT 06515
United States
Jobby Jacob
School of Mathematical Sciences
Rochester Institute of Technology
Rochester, NY 14623
United States