Extremal problems on planar graphs without k edge-disjoint cycles

Abstract

In the 1960s, Erd\H{o}s 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 $k\leq4$. As pointed out by Bollob\'{a}s, it is very difficult for general $k$. Recently, Tait and Tobin [J. Combin. 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.