A path in an edge−colored graph is said to be a rainbow path if every edge in the path has different color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of vertices. The rainbow connection of a graph G, denoted by rc(G), is the smallest number of colors required to color the edges of graph such that the graph is rainbow connected. Given two arbitrary vertices u and v in G, a rainbow u−v geodesic in G is a rainbow u−v path of length d(u, v), where d(u, v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u−v geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by src(G), is the minimum number of colors needed to make G strongly rainbow connected. In this paper we determine the exact values of rc(G) and src(G) where G are F n and S m with n + 1 and 2m vertices, respectively.

The locating chromatic number of a graph is defined as the cardinality of a minimum resolving partition of the vertex set ( ) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in are not contained in the same partition class. In this case, the coordinate of a vertex V in is expressed in terms of the distances of V to all partition classes. This concept is a special case of the graph partition dimension notion. In this paper we investigate the locating chromatic number for two families of barbell graphs.

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Penelitan ini bertujuan menentukan dimensi metrik dari salah satu graf Fullerne yaitu graf Buckminsterfullerene yang memiliki 60 titik. Satu-satunya graf dengan dimensi metrik satu adalah graf lintasan Pn dengan n ≥ 2 sehingga tidak mungkin graf Buckminsterfullerene memiliki dimensi metrik satu. Dari hasil penelitian ini diperoleh bahwa dimensi metrik dari graf Buckminsterfullerene adalah tiga.Kata Kunci: Dimensi metrik, himpunan pemisah, graf Fullerene, graf Buckminsterfullerene

<p>For some ordered subset <span class="math"><em>W</em> = {<em>w</em>₁, <em>w</em>₂, ⋯, <em>w</em>_<em>t</em>}</span> of vertices in connected graph <span class="math"><em>G</em></span>, and for some vertex <span class="math"><em>v</em></span> in <span class="math"><em>G</em></span>, the metric representation of <span class="math"><em>v</em></span> with respect to <span class="math"><em>W</em></span> is defined as the <span class="math"><em>t</em></span>-vector <span class="math"><em>r</em>(<em>v</em>∣<em>W</em>) = {<em>d</em>(<em>v</em>, <em>w</em>₁), <em>d</em>(<em>v</em>, <em>w</em>₂), ⋯, <em>d</em>(<em>v</em>, <em>w</em>_<em>t</em>)}</span>. The set <span class="math"><em>W</em></span> is the resolving set of <span class="math"><em>G</em></span> if for every two vertices <span class="math"><em>u</em>, <em>v</em></span> in <span class="math"><em>G</em></span>, <span class="math"><em>r</em>(<em>u</em>∣<em>W</em>) ≠ <em>r</em>(<em>v</em>∣<em>W</em>)</span>. The metric dimension of <span class="math"><em>G</em></span>, denoted by <span class="math">dim(<em>G</em>)</span>, is defined as the minimum cardinality of <span class="math"><em>W</em></span>. Let <span class="math"><em>G</em></span> be a connected graph on <span class="math"><em>n</em></span> vertices. The thorn graph of <span class="math"><em>G</em></span>, denoted by <span class="math"><em>T</em><em>h</em>(<em>G</em>, <em>l</em>₁, <em>l</em>₂, ⋯, <em>l</em>_<em>n</em>)</span>, is constructed from <span class="math"><em>G</em></span> by adding <span class="math"><em>l</em>_<em>i</em></span> leaves to vertex <span class="math"><em>v</em>_<em>i</em></span> of <span class="math"><em>G</em></span>, for <span class="math"><em>l</em>_<em>i</em> ≥ 1</span> and <span class="math">1 ≤ <em>i</em> ≤ <em>n</em></span>. The subdivided-thorn graph, denoted by <span class="math"><em>T</em><em>D</em>(<em>G</em>, <em>l</em>₁(<em>y</em>₁), <em>l</em>₂(<em>y</em>₂), ⋯, <em>l</em>_<em>n</em>(<em>y</em>_<em>n</em>))</span>, is constructed by subdividing every <span class="math"><em>l</em>_<em>i</em></span> leaves of the thorn graph of <span class="math"><em>G</em></span> into a path on <span class="math"><em>y</em>_<em>i</em></span> vertices. In this paper the metric dimension of thorn of complete graph, <span class="math">dim(<em>T</em><em>h</em>(<em>K</em>_<em>n</em>, <em>l</em>₁, <em>l</em>₂, ⋯, <em>l</em>_<em>n</em>))</span>, <span class="math"><em>l</em>_<em>i</em> ≥ 1</span> are determined, partially answering the problem proposed by Iswadi et al . This paper also gives some conjectures for the lower bound of <span class="math">dim(<em>T</em><em>h</em>(<em>G</em>, <em>l</em>₁, <em>l</em>₂, ⋯, <em>l</em>_<em>n</em>))</span>, for arbitrary connected graph <span class="math"><em>G</em></span>. Next, the metric dimension of subdivided-thorn of complete graph, <span class="math">dim(<em>T</em><em>D</em>(<em>K</em>_<em>n</em>, <em>l</em>₁(<em>y</em>₁), <em>l</em>₂(<em>y</em>₂), ⋯, <em>l</em>_<em>n</em>(<em>y</em>_<em>n</em>))</span> are determined and some conjectures for the lower bound of <span class="math">dim(<em>T</em><em>h</em>(<em>G</em>, <em>l</em>₁(<em>y</em>₁), <em>l</em>₂(<em>y</em>₂), ⋯, <em>l</em>_<em>n</em>(<em>y</em>_<em>n</em>))</span> for arbitrary connected graph <span class="math"><em>G</em></span> are given.</p>