The rainbow numbers of cycles in maximal bipartite planar graph (Q6585243)

Given a family of graphs \(\mathcal{H}\), a graph is \(\mathcal{H}\)-free if it contains no copy of a graph in \(\mathcal{H}\). Given an edge-colored graph \(G\), a subgraph \(H\) of \(G\) is rainbow if no pair of edges of \(H\) receive the same color. Given a not \(\mathcal{H}\)-free graph \(G\), the rainbow number of \(\mathcal{H}\) in \(G\), denoted as \(\operatorname{rb}(G,\mathcal{H})\), is the minimum integer \(t\) such that any \(t\)-edge-coloring of \(G\) produces a rainbow copy of some graph in \(\mathcal{H}\). Let \(\mathcal{B}_n\) denote the family of all maximal bipartite planar graphs on \(n\) vertices and let \(\mathcal{B}_n(\mathcal{H})\) denote the family of all the maximal bipartite planar graphs on \(n\) vertices which are not \(\mathcal{H}\)-free. The rainbow number of a family \(\mathcal{H}\) in maximal bipartite planar graph, denoted by \(\operatorname{rb}(\mathcal{B}_n(\mathcal{H}),\mathcal{H})\), is defined as \(\max\{\operatorname{rb}(G,\mathcal{H})|G\in \mathcal{B}_n(\mathcal{H})\}\). Let \(C_l\) be the cycle of order \(l\). The graph obtained by arbitrarily adding one pendant edge at one vertex of a cycle \(C_l\) is denoted by \(C^+_l\). For any positive integer \(k\), let \(\mathcal{C}_k= \{C_4, C_6, \dots, C_{2k+2}\}\) and \(\mathcal{C}^+_{k,2i_0}=\Big(\mathcal{C}_k\setminus \{C_{2i_o}\} \Big)\cup \{C^+_{2i_0}\} \) for some \(i_0\in \{2, 3, \dots, k+1\}\). In this paper the authors prove that \(\operatorname{rb}(\mathcal{B}_n, \mathcal{C}_k) = \operatorname{rb}(\mathcal{B}_n\), \(\mathcal{C}^+_{k,2i_0}) = n+\lfloor\frac{n-2}{k+2}\rfloor -1 \) for any \(n\geq \max \{2k+2, 5\}\) and \(k+1\geq i_0\geq 2\); and that for \(H\in \{C_{2k}, C_{2k}^+\}\), with \(k\geq 3\), \(\operatorname{rb}(\mathcal{B}_n(\{H\}), \{H\}) = 2n-4\) for any \(n\geq |V(H)|\).