A vertex z in a connected graph G resolves two vertices u and v in G if d G (u, z) ̸ = d G (v, z). A set of vertices R G {u, v} is a set of all resolving vertices of u and v in G. For every two distinct vertices u and v in G, a resolving function f of G is a real function f : V (G) → [0, 1] such that f (R G {u, v}) ≥ 1. The minimum value of f (V (G)) from all resolving functions f of G is called the fractional metric dimension of G. In this paper, we consider a graph which is obtained by the comb product between two connected graphs G and H, denoted by G £o H. For any connected graphs G, we determine the fractional metric dimension of G £o H where H is a connected graph having a stem or a major vertex.
Let K l×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G 1 and G 2 , and integer j ≥ 2, the size multipartite Ramsey number m j (G 1 , G 2 ) is the smallest integer t such that every factorization of the graph K j×t := F 1 ⊕ F 2 satisfies the following condition: either F 1 contains G 1 or F 2 contains G 2 . In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths P n versus stars, for n = 2, 3 only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths P n versus stars, for n = 3, 4, 5, 6. In this paper, we investigate the size tripartite Ramsey numbers of paths P n versus stars, with all n ≥ 2. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers m 2 (K 1,m , C n ) of stars versus cycles, for n ≥ 3, m ≥ 2.
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