Extremal problems on planar graphs without k edge-disjoint cycles

TL;DR

This paper investigates the maximum number of edges and spectral radius in planar graphs without k edge-disjoint cycles, extending classical extremal graph theory results and providing new bounds and unique extremal graphs.

Contribution

It determines the maximum edges and spectral radius for such planar graphs, introducing new extremal bounds and characterizing the unique extremal graphs for these properties.

Findings

Maximum edges in planar graphs without k edge-disjoint cycles identified.

Maximum spectral radius for these graphs established.

Unique extremal graphs characterized.

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.