On the connected coalition number of graphs

<p>For a graph <span class="math inline">\(G=(V,E)\)</span>, a pair of vertex disjoint sets <span class="math inline">\(A_{1}\)</span> and <span class="math inline">\(A_{2}\)</span> form a connected coalition of <span class="math inline">\(G\)</span>, if <span class="math inline">\(A_{1}\cup A_{2}\)</span> is a connected dominating set, but neither <span class="math inline">\(A_{1}\)</span> nor <span class="math inline">\(A_{2}\)</span> is a connected dominating set. A connected coalition partition of <span class="math inline">\(G\)</span> is a partition <span class="math inline">\(\Phi\)</span> of <span class="math inline">\(V(G)\)</span> such that each set in <span class="math inline">\(\Phi\)</span> either consists of only a singe vertex with the degree <span class="math inline">\(\mid V(G)\mid-1\)</span>, or forms a connected coalition of <span class="math inline">\(G\)</span> with another set in <span class="math inline">\(\Phi\)</span>. The connected coalition number of <span class="math inline">\(G\)</span>, denoted by <span class="math inline">\(CC(G)\)</span>, is the largest possible size of a connected coalition partition of <span class="math inline">\(G\)</span>. In this paper, we characterize graphs that satisfy <span class="math inline">\(CC(G)=2\)</span>. Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.</p>