Extremal problems on planar graphs without k edge-disjoint cycles

Abstract

In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on n vertices without k edge-disjoint cycles. This problem had been solved for $k\le 4$. As pointed out by Bollobás, it is very difficult for general k. Recently, Tait and Tobin [J. Comb. Theory, Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of n-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without k edge-disjoint cycles. We first determine the maximum number of edges in a planar graph of order n and maximum degree $n-1$ without k edge-disjoint cycles. Based on this, we then determine the maximum spectral radius as well as its unique extremal graph over all planar graphs on n vertices without k edge-disjoint cycles. Finally, we also discuss several extremal problems for general graphs.