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Radio Graceful Hamming Graphs

100%

Niedzialomski A.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 4

1007-1020

For k ∈ ℤ+ and G a simple, connected graph, a k-radio labeling f : V (G) → ℤ+ of G requires all pairs of distinct vertices u and v to satisfy |f(u) − f(v)| ≥ k + 1 − d(u, v). We consider k-radio labelings of G when k = diam(G). In this setting, f is injective; if f is also surjective onto {1, 2, . . . , |V (G)|}, then f is a consecutive radio labeling. Graphs that can be labeled with such a labeling are called radio graceful. In this paper, we give two results on the existence of radio graceful Hamming graphs. The main result shows that the Cartesian product of t copies of a complete graph is radio graceful for certain t. Graphs of this form provide infinitely many examples of radio graceful graphs of arbitrary diameter. We also show that these graphs are not radio graceful for large t.

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Radio numbers for generalized prism graphs

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Martinez P., Ortiz J., Tomova M., Wyels C.

Discussiones Mathematicae Graph Theory

2011

tom 31

nr 1

45-62

A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted $Z_{n,s}$, s ≥ 1, n ≥ s, have vertex set {(i,j) | i = 1,2 and j = 1,...,n} and edge set {((i,j),(i,j ±1))} ∪ {((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋}. In this paper we determine the radio number of $Z_{n,s}$ for s = 1,2 and 3. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

1 Martinez P.

1 Niedzialomski A.

1 Ortiz J.

1 Tomova M.

1 Wyels C.

1 2016

1 2011

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Radio Graceful Hamming Graphs

Radio numbers for generalized prism graphs