论文标题
没有k边缘循环的平面图上的极端问题
Extremal problems on planar graphs without k edge-disjoint cycles
论文作者
论文摘要
在1960年代，Erdős和他的合作者在没有$ n $ n $顶点的图表或平面图中启动了最大边缘数量的研究。此问题已以$ k \ leq4 $解决。正如Bollobás指出的那样，对于一般$ k $来说，这是非常困难的。最近，Tait和Tobin [J。组合。理论ser。 B，2017年]确认了$ n $ vertex Planar图的最大光谱半径上的著名猜想。在上述结果的激励下，我们考虑了在没有$ K $ edge-disschoint周期的平面图上的两个极端问题。我们首先确定订单$ n $的平面图和最高度$ n-1 $的最大边数，而没有$ k $ edge-edise-exch-dischoint Cycles。基于此，我们然后确定最大光谱半径以及其在$ n $顶点上所有平面图上的独特极端图，而没有$ k $ edge-edise-disschincles Cycles。最后，我们还讨论了一些通用图的极端问题。
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 $k\leq4$. As pointed out by Bollobás, 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.
