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We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph.
We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order.
The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)leq 3$. In two recent papers, it was proved that planar gr
We prove a `resilience version of Diracs theorem in the setting of random regular graphs. More precisely, we show that, whenever $d$ is sufficiently large compared to $varepsilon>0$, a.a.s. the following holds: let $G$ be any subgraph of the random $
In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.
All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constr