An edge-colored graph G is called a rainbow connected if any two vertices are connected by a path whose edges have distinct colors. Such a path is called a rainbow path. The smallest number of colors required in order to make G rainbow connected is called the rainbow connection number of G. For two connected graphs G and H with v ∈ V (H), the comb product between G and H, denoted by G ▷ v H, is a graph obtained by taking one copy of G and |V (G)| copies of H and identifying the i-th copy of H at the vertex v to the i-th vertex of G. In this paper, we give sharp lower and upper bounds for the rainbow connection number of comb product between two connected graphs. We also determine the exact values of rainbow connection number of G ▷ v H for some connected graphs G and H.
A set <em>W</em> is called a local resolving set of <em>G</em> if the distance of <em>u</em> and <em>v</em> to some elements of <em>W</em> are distinct for every two adjacent vertices <em>u</em> and <em>v</em> in <em>G</em>. The local metric dimension of <em>G</em> is the minimum cardinality of a local resolving set of <em>G</em>. A connected graph <em>G</em> is called a split graph if <em>V</em>(<em>G</em>) can be partitioned into two subsets <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub> where an induced subgraph of G by <em>V</em><sub>1</sub> and <em>V</em><sub>2</sub> is a complete graph and an independent set, respectively. We also consider a graph, namely the unicyclic graph which is a connected graph containing exactly one cycle. In this paper, we provide a general sharp bounds of local metric dimension of split graph. We also determine an exact value of local metric dimension of any unicyclic graphs.
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