On Radio Number of Stacked-Book Graphs
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Date
2019
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arXiv preprint arXiv
Abstract
A Stacked-book graph Gm,n results from the Cartesian product of a star graph Sm and path Pn, where m and n are the orders of Sm and Pn respectively. A radio labeling problem of a simple and connected graph, G, involves a non- negative integer function f : V (G) → Z+ on the vertex set V (G) of G, such that for all u, v ∈ V (G), |f(u) − f(v)| ≥ diam(G) + 1 − d(u, v), where diam(G) is the diameter of G and d(u, v) is the shortest distance between u and v. Suppose that fmin and fmax are the respective least and largest values of f on V (G), then, spanf, the absolute difference of fmin and fmax, is the span of f while the radio number rn(G) of G is the least value of spanf over all the possible radio labels on V (G). In this paper, we obtain the radio number for the stacked-book graph Gm,n where m ≥ 4 and n is even, and obtain bounds for m = 3 which improves existing upper and lower bounds for Gm,n where m = 3.
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Adefokun, T. C. and Ajayi, D. O. (2019). On Radio Number of Stacked-Book Graphs. arXiv preprint arXiv. 19(1).