Optimal radio labelings of graphs
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Let ℕ be the set of positive integers. A radio labeling of a graph G is a mapping φ : V(G) →ℕ∪{0} such that the inequality |φ(u)-φ(v)| ≥ diam(G) + 1 - d(u,v) holds for every pair of distinct vertices u,v of G, where diam(G) and d(u,v) are the diameter of G and distance between u and v in G, respectively. The radio number rn(G) of G is the smallest number k such that G has radio labeling φ with max{φ(v) : v ∈ V(G)} = k. Das et al. [Discrete Math. 340(2017) 855-861] gave a technique to find a lower bound for the radio number of graphs. In [Algorithms and Discrete Applied Mathematics: CALDAM 2019, Lecture Notes in Computer Science 11394, springer, Cham, 2019, 161-173], Bantva modified this technique for finding an improved lower bound on the radio number of graphs and gave a necessary and sufficient condition to achieve the improved lower bound. In this paper, one more useful necessary and sufficient condition to achieve the improved lower bound for the radio number of graphs is given. Using this result, the radio number of the Cartesian product of a path and a wheel graphs is determined.
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